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Example 2 Contractible Sullivan algebras have contractible realizations.
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Suppose \( A = \left( {\Lambda \left( {U \oplus {dU}}\right), d}\right) \) is a contractible Sullivan algebra, and let \( \left\{ {u}_{\alpha }\right\} \) be a basis of \( U \) . An element of \( \langle A{\rangle }_{n} \) is a morphism \( \varphi : A \rightarrow {\left( {A}_{PL}\right) }_{n} \), and so an isomorphism  is given by \( \varphi \mapsto \left\{ {\varphi {u}_{\alpha }}\right\} \) . By Lemma 10.7 each \( {\left( {A}_{PL}\right) }_{\left| {u}_{\alpha }\right| } \) is extendable, and hence so is the product (a trivial exercise). It follows (Example 3, \( §{17} \) (a)) that \( \left| A\right| \) is contractible.
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No
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Theorem 17.10 Let \( \left( {{\Lambda V}, d}\right) \) be a Sullivan algebra such that \( {H}^{1}\left( {{\Lambda V}, d}\right) = 0 \) and each \( {H}^{p}\left( {{\Lambda V}, d}\right) \) is finite dimensional. Then\n\n(i) \( \left| {{\Lambda V}, d}\right| \) is simply connected and \( {\zeta }_{n} : {\pi }_{n}\left( \left| {{\Lambda V}, d}\right| \right) \overset{ \cong }{ \rightarrow }{\operatorname{Hom}}_{Ik}\left( {V,\mathbb{k}}\right) \) is an isomorphism, \( n \geq 2 \).\n\n(ii) If \( \mathbb{k} = \mathbb{Q} \) then \( {m}_{\left( \Lambda V, d\right) } \) is a quasi-isomorphism,\n\n\[ {m}_{\left( \Lambda V, d\right) } : \left( {{\Lambda V}, d}\right) \overset{ \simeq }{ \rightarrow }{A}_{PL}\left( \left| {{\Lambda V}, d}\right| \right) . \]
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proof of Theorem 17.10: We first reduce to the case \( \left( {{\Lambda V}, d}\right) \) is minimal. Use Theorem 14.9 to write \( \left( {{\Lambda V}, d}\right) = \left( {{\Lambda W}, d}\right) \otimes \left( {\Lambda \left( {U \oplus {dU}}\right), d}\right) \) with \( \left( {{\Lambda W}, d}\right) \) minimal. Then \( \left| {{\Lambda V}, d}\right| \) is the product of \( \left| {{\Lambda W}, d}\right| \) and a contractible CW complex (Examples 1 and 2, \( §{17}\left( \mathrm{c}\right) \) ). It is thus sufficient to prove the theorem for \( \left( {{\Lambda W}, d}\right) \) . We may therefore assume that \( \left( {{\Lambda V}, d}\right) \) itself is minimal. This implies that \( V = \) \( {\left\{ {V}^{p}\right\} }_{p > 2} \) and that each \( {V}^{p} \) is finite dimensional.\n\nWe shall rely on the following observation. Suppose \( \left( {{\Lambda U}, d}\right) \rightarrow \left( {{\Lambda U} \otimes {\Lambda V}, d}\right) \) is a minimal relative Sullivan algebra in which \( \left( {{\Lambda U}, d}\right) \) is itself a minimal Sullivan\n\nalgebra. As in Proposition 17.9 this realizes to a fibre bundle \( \left| {{\Lambda U} \otimes {\Lambda V}, d}\right| \overset{\left| \lambda \right| }{ \rightarrow } \) \( \left| {{\Lambda U}, d}\right| \) with fibre \( \left| {{\Lambda V},\bar{d}}\right| \) . In particular, \( \left| \lambda \right| \) is a Serre fibration (Proposition 2.6). Let \( {\partial }_{ * } : {\pi }_{ * }\left( \left| {{\Lambda U}, d}\right| \right) \rightarrow {\pi }_{* - 1}\left( \left| {{\Lambda V},\bar{d}}\right| \right) \) be the connecting homeomorphism.\n\nNext, by adjointness we have the commutative diagram\n\n\n\n\n\nUse the Lifting lemma 14.4 to choose \( {m}_{\left( \Lambda U, d\right) },{m}_{\left( \Lambda U \otimes \Lambda V, d\right) } \) and \( {m}_{\left( \Lambda V,\bar{d}\right) } \) so that the diagram \n\ncommutes.\n\nNow the argument of Proposition 15.13 establishes that\n\n\[ \left\langle {{d}_{0}v;\alpha }\right\rangle = {\left( -1\right) }^{n + 1}\left\langle {v;{\partial }_{ * }\alpha }\right\rangle ,\;v \in {V}^{n},\alpha \in {\pi }_{n + 1}\left( \left| {\Lambda U}\right| \right) . \]\n\nThis translates to the commutative diagram\n\n\n\n(17.11)\n\nin which \( {d}_{0}^{ * } \) is the dual of \( {d}_{0} : \left( {{d}_{0}^{ * }f}\right) \left( v\right) = {\left( -1\right) }^{n + 1}f\left( {{d}_{0}v}\right), v \in {V}^{n} \).
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Yes
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Theorem 17.12 With the notation and hypotheses above,\n\n(i) The diagram\n\n\n\n is homotopy commutative.\n\n(ii) If \( \\mathbf{k} = \\mathbb{Q} \) then all the morphisms in the diagram are quasi-isomorphisms. In particular, \( {h}_{X} \) is a rationalization of \( X\\left( {§9\\left( b\\right) }\\right) \) .
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proof: (i) The left hand triangle commutes by construction. Next observe that \( {S}_{ * }\\left( {s}_{X}\\right) \\circ {\\xi }_{{S}_{ * }\\left( X\\right) } = {id} : {S}_{ * }\\left( X\\right) \\rightarrow {S}_{ * }\\left( X\\right) \) . Since \( {t}_{X}{s}_{X} \\sim i{d}_{\\left| SX\\right| } \) we can apply Proposition 12.6 to conclude that\n\n\\[ \n\\begin{matrix} {A}_{PL}\\left( {\\xi }_{{S}_{ * }\\left( X\\right) }\\right) \\circ \\varphi & \\sim & {A}_{PL}\\left( {\\xi }_{{S}_{ * }\\left( X\\right) }\\right) \\circ {A}_{PL}\\left( {s}_{X}\\right) \\circ {A}_{PL}\\left( {t}_{X}\\right) \\circ \\varphi \\end{matrix} \n\\]\n\n\\[ \n= {A}_{PL}\\left( {t}_{X}\\right) \\circ \\varphi \n\\]\n\nfor any morphism \( \\varphi : \\left( {{\\Lambda W, d}}\\right) \\rightarrow {A}_{PL}\\left( \\left| {{S}_{ * }\\left( X\\right) }\\right| \\right) \) . Thus\n\n\\[ \n\\begin{matrix} {A}_{PL}\\left( {h}_{X}\\right) \\circ {m}_{\\left( \\Lambda W, d\\right) } & \\sim & {A}_{PL}\\left( {\\xi }_{{S}^{ * }\\left( X\\right) }\\right) \\circ {A}_{PL}\\left( \\left| {\\gamma }_{X}\\right| \\right) \\circ {m}_{\\left( \\Lambda W, d\\right) } \\end{matrix} \n\\]\n\n\\[ \n= {A}_{PL}\\left( {\\gamma }_{X}\\right) \\circ {A}_{PL}\\left( {\\xi }_{\\langle {\\Lambda W, d}\\rangle }\\right) \\circ {m}_{\\left( \\Lambda W, d\\right) } \n\\]\n\n\\[ \n= {A}_{PL}\\left( {\\gamma }_{X}\\right) \\circ {\\eta }_{\\langle {\\Lambda W, d}\\rangle } \n\\]\n\n\\[ \n= {m}_{X}\\text{,} \n\\]\n\nthe last equality following from the adjointness of \( {\\gamma }_{X} \) and \( {m}_{X} \) .\n\n(ii) According to Theorem 17.10, \( {m}_{\\left( \\Lambda W, d\\right) } \) is a quasi-isomorphism. Hence so is \( {A}_{PL}\\left( {h}_{X}\\right) \) . In \( §{17}\\left( \\mathrm{a}\\right) \) we observed that \( {C}_{ * }\\left( {\\xi }_{K}\\right) \) is a quasi-isomorphism for any simplicial set \( K \) . In particular \( {A}_{PL}\\left( {\\xi }_{\\langle {\\Lambda W, d}\\rangle }\\right) \) is a quasi-isomorphism. Hence so is \( {\\eta }_{\\left( \\Lambda W, d\\right) } \).\n\nFinally since \( \\left| {{\\Lambda W, d}}\\right| \) is a simply connected rational space and since \( {H}_{ * }\\left( {{h}_{X};\\mathbb{Q}}\\right) \) is an isomorphism (because \( {A}_{PL}\\left( {h}_{X}\\right) \) is a quasi-isomorphism) it follows that \( {h}_{X} \) is a rationalization (Theorem 9.6).
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Yes
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Proposition 17.13 If \( {\varphi }_{0} \sim {\varphi }_{1} : \left( {{\Lambda V}, d}\right) \rightarrow \left( {{\Lambda W}, d}\right) \) are homotopic morphisms between Sullivan algebras then
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proof: A homotopy from \( {\varphi }_{0} \) to \( {\varphi }_{1} \) is a morphism \( \Phi : \left( {{\Lambda V}, d}\right) \rightarrow \left( {{\Lambda W}, d}\right) \otimes \) \( \Lambda \left( {t,{dt}}\right) \) such that \( \left( {{id} \otimes {\varepsilon }_{i}}\right) \Phi = {\varphi }_{i} \), where \( {\varepsilon }_{i}\left( t\right) = i \) . Since realization preserves products (the Example of \( §{17}\left( \mathrm{c}\right) \) ) we obtain a continuous map \( \left| \Phi \right| : \left| {{\Lambda W}, d}\right| \times \) \( \left| {\Lambda \left( {t,{dt}}\right) }\right| \rightarrow \left| {{\Lambda V}, d}\right| \) such \( \left| \Phi \right| \left( {x,{\varepsilon }_{i}}\right) = \left| {\varphi }_{i}\right| \left( x\right) \) . But the identity map of \( \Lambda \left( {t,{dt}}\right) \) is a 1-cell in \( \left| {\Lambda \left( {t,{dt}}\right) }\right| \) joining \( {\varepsilon }_{0} \) to \( {\varepsilon }_{1} \), and so \( \left| {\varphi }_{0}\right| \sim \left| {\varphi }_{1}\right| \) .
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Yes
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Theorem 17.15 With the hypothesis and notation above, let \( \varphi : \left( {{\Lambda V}, d}\right) \rightarrow \) \( \left( {{\Lambda W}, d}\right) \) be a Sullivan representative for \( f \) . Then the diagram\n\n\n\n is homotopy commutative.
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proof: In the notation at the end of \( §{17}\left( \mathrm{\;d}\right) ,{t}_{X} \) and \( {t}_{Y} \) are homotopy inverses to \( {s}_{X} \) and \( {s}_{Y} \) . Since \( f{s}_{X} = {s}_{Y}\left| {{S}_{ * }\left( f\right) }\right| \) it follows that \( {t}_{Y}f \sim \left| {{S}_{ * }\left( f\right) }\right| {t}_{X} \) . But \( {h}_{X} = \left| {\gamma }_{X}\right| \circ {t}_{X} \), and so it is sufficient to prove that \( \left| {\gamma }_{Y}\right| \circ \left| {{S}_{ * }\left( f\right) }\right| \sim \left| \varphi \right| \circ \left| {\gamma }_{X}\right| \) .\n\nBecause \( \varphi \) is a Sullivan representative of \( f \) there is a cochain algebra morphism \( \Psi : \left( {{\Lambda V}, d}\right) \rightarrow {A}_{PL}\left( X\right) \otimes \Lambda \left( {t,{dt}}\right) \) such that \( \left( {{id} \otimes {\varepsilon }_{0}}\right) \Psi = {A}_{PL}\left( f\right) {m}_{Y} \) and \( \left( {{id} \otimes {\varepsilon }_{1}}\right) \Psi = {m}_{X}\varphi \) . Compose this with the obvious morphism \( {A}_{PL}\left( X\right) \otimes \) \( \Lambda \left( {t,{dt}}\right) \rightarrow {A}_{PL}\left( {{S}_{ * }\left( X\right) \times \Delta \left\lbrack 1\right\rbrack }\right) \) and take the adjoint morphism \( \omega : {S}_{ * }\left( X\right) \times \) \( \Delta \left\lbrack 1\right\rbrack \rightarrow \langle {\Lambda V}, d\rangle \) via the adjoint relation (7.8). Then \( \left| \omega \right| : \left| {{\gamma }_{Y}{S}_{ * }\left( f\right) }\right| \sim \left| {\varphi {\gamma }_{X}}\right| \) .
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Yes
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Proposition 17.16 Suppose \( \left( {{\Lambda V}, d}\right) \) is a Sullivan algebra in which each \( {V}^{n} \) is finite dimensional and either \( V = {V}^{ \geq 2} \) or else \( d \) preserves \( V \) . Then \( {\int }_{ * } \) is a quasi-isomorphism.
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proof: Theorem 10.15(ii) asserts that \( \oint \) is a quasi-isomorphism. On the other hand, so is any canonical Sullivan model, \( {m}_{\left( \Lambda V, d\right) } \) . (If \( V = {V}^{ \geq 2} \) this is Theorem 17.10. If \( d : V \rightarrow V \) a trivial modification of the proof gives the same result.) Since \( \eta = {A}_{PL}\left( \xi \right) \circ {m}_{\left( \Lambda V, d\right) } \) it follows that \( \eta \) is a quasi-isomorphism too (§17(d)). Thus the composite dualizes to a quasi-isomorphism \( \operatorname{Hom}\left( {{C}^{ * }\langle {\Lambda V}, d\rangle ,\mathbb{R}}\right) \overset{ \simeq }{ \rightarrow } \) \( {C}_{\left( \Lambda V, d\right) } \) . Finally, if \( C \) is any chain complex there is a natural morphism \( C \rightarrow \operatorname{Hom}\left( {\operatorname{Hom}\left( {C,\mathbb{k}}\right) ,\mathbb{k}}\right) \) . If \( C \) itself has finite type this is an isomorphism. If \( H\left( C\right) \) has finite type then this is a quasi-isomorphism because \( H\left( {\operatorname{Hom}\left( {-,\mathbb{R}}\right) }\right) = \) \( \operatorname{Hom}\left( {H\left( -\right) ,\mathbb{R}}\right) \) . In the case of \( \left( {{\Lambda V}, d}\right) \) we have \( {H}^{ * }\langle {\Lambda V}, d\rangle \cong H\left( {{\Lambda V}, d}\right) \) and so \( {H}_{ * }\langle {\Lambda V}, d\rangle \) has finite type. Thus since \( {C}^{ * }\langle {\Lambda V}, d\rangle = \operatorname{Hom}\left( {{C}_{ * }\langle {\Lambda V}, d\rangle ,\mathbb{k}}\right) \) the inclusion \( \lambda : {C}_{ * }\langle {\Lambda V}, d\rangle \rightarrow \operatorname{Hom}\left( {{C}^{ * }\langle {\Lambda V}, d\rangle ,\mathbb{R}}\right) \) is a quasi-isomorphism too. Hence so is the composite \( {\int }_{ * } = \operatorname{Hom}\left( {\oint \circ \eta ,\mathbb{k}}\right) \circ \lambda \) .
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Yes
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There is one particular situation that arises frequently in practice. We suppose given a graded differential module \( \left( {M, D}\right) \) in which\n\n- Each \( {M}^{r} \) is given as a direct sum \( {M}^{r} = {\bigoplus }_{p + q = r}{M}^{p, q} \) .\n\n- The differential \( d \) is a direct sum \( D = \mathop{\sum }\limits_{{i \geq 0}}{D}_{i} \) with \( {D}_{i} : {M}^{p, q} \rightarrow {M}^{p + i, q - i + 1} \) .\n\nHere a canonical filtration is given by \( {F}^{k}{\left( M\right) }^{r} = {\bigoplus }_{p \geq k}{M}^{p, r - p} \) and this leads as just described to a spectral sequence.
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Moreover from \( {D}^{2} = 0 \) we deduce \( {D}_{0}^{2} = 0,{D}_{1}{D}_{0} + {D}_{0}{D}_{1} = 0 \) and \( {D}_{1}^{2} = \) \( - {D}_{0}{D}_{2} + {D}_{2}{D}_{0} \) . In particular, \( {D}_{0} \) is itself a differential and \( {D}_{1} \) induces a differential \( H\left( {D}_{1}\right) \) in \( H\left( {M,{D}_{0}}\right) \) . Now a straightforward verification shows that the first two terms of the spectral sequence are given by\n\n\[ \left( {{E}_{0},{d}_{0}}\right) = \left( {M,{D}_{0}}\right) \;\text{ and }\;\left( {{E}_{1},{d}_{1}}\right) = \left( {H\left( {M,{D}_{0}}\right), H\left( {D}_{1}}\right) }\right) .\n\]
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Yes
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Proposition 18.1 Suppose \( \\left( {M, d,\\mathfrak{F}}\\right) \) is a cochain complex with a first quadrant filtration. Then the associated spectral sequence is first quadrant, and converges to \( H\\left( M\\right) \) .
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proof: The first assertion is immediate since \( {E}_{0} = \\mathcal{G}M \) is necessarily concentrated in non-negative bidegrees.\n\nFor the second assertion let \( {Z}_{r}^{p, q} \) be the component of degree \( p + q \) in the graded module \( {Z}_{r}^{p} \) . If \( r > q + 1 \) then\n\n\[ d : {Z}_{r}^{p, q} \\rightarrow {\\left( {F}^{p + r}M\\right) }^{p + q + 1} = 0. \]\n\nThus\n\n\( {Z}_{r}^{p, q} = \\ker d \\cap {\\left( {F}^{p}M\\right) }^{p + q}\\; \) and, similarly, \( \\;{Z}_{r - 1}^{p + 1, q - 1} = \\ker d \\cap {\\left( {F}^{p + 1}M\\right) }^{p + q}. \)\n\nMoreover, if \( r > p \) then \( {D}_{r - 1}^{p} = d\\left( M\\right) \\cap {F}^{p}M \) . This identifies\n\n\[ {E}_{r}^{p, q} = {Z}_{r}^{p, q}/\\left( {{Z}_{r - 1}^{p + 1, q - 1} + {\\left( {D}_{r - 1}^{p}\\right) }^{p + q}}\\right) \\cong {\\mathcal{G}}^{p, q}H\\left( M\\right) . \]
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Yes
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Proposition 18.2 (Comparison) Let \( \varphi : \left( {M, d,\mathfrak{F}}\right) \rightarrow \left( {N, d,\mathfrak{F}}\right) \) be a morphism of filtered cochain complexes with first quadrant filtrations. If some \( {E}_{r}\left( \varphi \right) \) is a quasi-isomorphism then \( H\left( \varphi \right) \) is an isomorphism of filtered modules. In particular, \( \varphi \) is a quasi-isomorphism.
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proof of 18.2: Since \( {E}_{i + 1}\left( \varphi \right) \) is identified with \( H\left( {{E}_{i}\left( \varphi \right) }\right) \) it follows that \( {E}_{r + 1}\left( \varphi \right) \) is an isomorphism. By induction \( {E}_{m}\left( \varphi \right) \) is an isomorphism for \( m \geq r + 1 \) . Hence \( {E}_{\infty }\left( \varphi \right) \) is an isomorphism. By Proposition 18.1 this is identified with \( \mathcal{G}H\left( \varphi \right) \) . Now apply the Lemma above (and its proof) with \( \psi = H\left( \varphi \right) \) .
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Yes
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Lemma 18.3 Suppose \( \psi : \left( {M,\mathfrak{F}}\right) \rightarrow \left( {N,\mathfrak{F}}\right) \) is a morphism of filtered graded modules with first quadrant (cohomology) filtrations. If \( \mathcal{G}\left( \psi \right) \) is injective (resp. surjective) then so is \( \psi \) .
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proof: Suppose \( \mathcal{G}\left( \psi \right) \) is injective. If \( 0 \neq x \in {M}^{n} \) there is a greatest \( p \) such that \( x \in {F}^{p}M \), because \( {\left( {F}^{n + 1}M\right) }^{n} = 0 \) . Thus \( x \) represents a non-zero element \( \left\lbrack x\right\rbrack \in {\mathcal{G}}^{p, n - p}\left( M\right) \) and so \( 0 \neq \mathcal{G}\left( \psi \right) \left\lbrack x\right\rbrack = \left\lbrack {\psi x}\right\rbrack \) . In particular \( {\psi x} \neq 0 \) .\n\nSuppose \( \mathcal{G}\left( \psi \right) \) is surjective. Then for each \( p,\psi \left( {{F}^{p}M}\right) + {F}^{p + 1}N = {F}^{p}N \) . It follows by induction on \( q \) that \( \psi \left( {{F}^{p}M}\right) + {F}^{p + q}N = {F}^{p}N, q \geq 0 \) . Fix \( n \) and choose \( q \) so \( p + q > n \) . Then \( {\left( {F}^{p + q}N\right) }^{n} = 0 \) . It follows that \( \psi \left( {{F}^{p}M}\right) = {F}^{p}N \) . \( ▱ \)
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Yes
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Proposition 19.1 With the notation above, \( \varrho \) is a quasi-isomorphism.
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proof: Since \( \varrho \) is surjective we need only show \( H\left( {\ker \varrho }\right) = 0 \) . Define \( h \) : \( B\left( {TV}\right) \rightarrow \ker \varrho \) by \( h\left( 1\right) = 0 \) and\n\n\[ h : \left\{ \begin{array}{l} \left\lbrack {{sv} \mid \cdots }\right\rbrack \rightarrow 0,\;v \in V \\ \left\lbrack {s\left( {{v}_{1} \otimes \cdots \otimes {v}_{\ell }}\right) \mid \cdots }\right\rbrack \mapsto {\left( -1\right) }^{\deg {v}_{1} + 1}\left\lbrack {s{v}_{1}\left| {s\left( {{v}_{2} \otimes \cdots \otimes {v}_{\ell }}\right) }\right| \cdots }\right\rbrack ,\;\ell \geq 2. \end{array}\right. \]\n\nA quick calculation shows that \( h{d}_{1} + {d}_{1}h = {id} \) in ker \( \varrho \) . Since \( h \) raises wordlength and \( {d}_{0} \) preserves it, it follows that \( \left( {{id} - {hd} - {dh}}\right) \) increases wordlength in \( \ker \varrho \) . Similarly, \( {id} - {hd} - {dh} \) preserves degrees.\n\nLet \( z \in \ker \varrho \) be a cycle of degree \( n \) . Then \( {\left( id - hd - dh\right) }^{n}z \) has degree \( n \) and wordlength at least \( n + 1 \) . However, our hypothesis on \( V \) implies that elements in \( {B}^{ \geq n + 1}\left( {TV}\right) \) have degree at least \( n + 1 \) . Thus \( {\left( id - hd - dh\right) }^{n}z = 0 \) . Since \( d{\left( id - hd - dh\right) }^{k}z = 0 \) this gives \( {\left( id - dh\right) }^{n}z = 0 \) ; i.e.\n\n\[ z = {dh}\left( {\mathop{\sum }\limits_{{i = 0}}^{{n - 1}}{\left( -1\right) }^{i}\left( \begin{matrix} n \\ i + 1 \end{matrix}\right) {\left( dh\right) }^{i}z}\right) \]\n\nis a boundary.
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Yes
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Lemma 20.1 Every A-module \( M \) has a free resolution.
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proof: Choose an \( A \) -linear surjection of degree zero, \( \varrho : {P}_{0, * } \rightarrow M \), from a free \( A \) -module \( {P}_{0, * } \) . If\n\n\[ M\overset{\varrho }{ \leftarrow }{P}_{0, * }\overset{d}{ \leftarrow }\cdots \overset{d}{ \leftarrow }{P}_{k, * } \]\n\n\nis constructed choose an \( A \) -linear surjection \( d \) of bidegree \( \left( {-1,0}\right) \) from a free \( A \) -module \( {P}_{k + 1, * } \) onto \( \ker d \subset {P}_{k, * } \) .
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No
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Lemma 20.2 Suppose the A-modules \( {P}_{k, * } \) are projective and \( \xi \) is a surjective quasi-isomorphism. Put \( r = \left( {p + q}\right) - \left( {m + n}\right) \). Then\n\n(i) There is a morphism \( \beta : \left( {P, d}\right) \rightarrow \left( {N, d}\right) \) of bidegree \( \left( {p - m, q - n}\right) \), and such that \( {\xi \beta } = \alpha \).\n\n(ii) If \( \widehat{\beta } \) is a second such morphism then \( \beta - \widehat{\beta } = {d\gamma } + {\left( -1\right) }^{r}{\gamma d} \), for some \( A \) -linear map \( \gamma : P \rightarrow \ker \xi \) of bidegree \( \left( {p - m + 1, q - n}\right) \).
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proof: (i) Suppose \( \beta \) constructed in \( {P}_{i, * }, i < k \) . Since \( {P}_{k, * } \) is projective there is an \( A \) -linear map \( {\beta }^{\prime } : {P}_{k, * } \rightarrow N \) such that \( \xi {\beta }^{\prime } = \alpha \) . Clearly \( d{\beta }^{\prime } - {\left( -1\right) }^{r}{\beta d} \) sends \( {P}_{k, * } \) into the cycles of \( \ker \xi \) . Since, by hypothesis, \( H\left( {\ker \xi }\right) = 0 \), it follows that \( d : \ker \xi \rightarrow \operatorname{cycles}\left( {\ker \xi }\right) \) is surjective. Choose \( {\beta }^{\prime \prime } : {P}_{k, * } \rightarrow \ker \xi \) so that \( d{\beta }^{\prime \prime } = d{\beta }^{\prime } - {\left( -1\right) }^{r}{\beta d} \) and set \( \beta = {\beta }^{\prime } - {\beta }^{\prime \prime } \) in \( {P}_{k, * } \).\n\n(ii) Note that \( \beta - \widehat{\beta } : \left( {P, d}\right) \rightarrow \left( {\ker \xi, d}\right) \). If \( \gamma \) is constructed in \( {P}_{i, * } \) , \( i < k \) then \( \beta - \widehat{\beta } - {\left( -1\right) }^{r}{\gamma d} \) sends \( {P}_{k, * } \) into the cycles of \( \ker \xi \) . Again, \( d : \ker \xi \rightarrow \) cycles \( \left( {\ker \xi }\right) \) is surjective, which permits us to construct \( \gamma : {P}_{k, * } \rightarrow \ker \xi \) so that \( {d\gamma } = \beta - \widehat{\beta } - {\left( -1\right) }^{r}{\gamma d}.
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Yes
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Proposition 20.10 For any A-module, \( M \) ,\n\n\[{\operatorname{projdim}}_{A}\left( M\right) = \sup \left\{ {k \mid {\operatorname{Ext}}_{A}^{k}\left( {M, - }\right) \text{ is non-zero }}\right\} .
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proof: Denote the right hand side of the equation by \( p\left( M\right) \) . If \( M \) has a projective resolution of the form above then we can use it to compute \( {\operatorname{Ext}}_{A}^{n}\left( {M, - }\right) \) and it follows at once that \( {\operatorname{Ext}}_{A}^{n}\left( {M, - }\right) = 0, n > k \) . Hence \( p\left( M\right) \leq \operatorname{proj}{\dim }_{A}\left( M\right) \) .\n\nConversely, suppose \( p\left( M\right) \) is finite and let \( \varrho : \left( {\left\{ {P}_{k, * }\right\}, d}\right) \rightarrow \left( {M,0}\right) \) be any projective resolution. For simplicity write \( p = p\left( M\right) \) . Let \( Z \subset {P}_{p, * } \) be the submodule of cycles \( \left( {Z = \ker d}\right) \) . Then the differential, \( d \), maps \( {P}_{p + 1, * } \) onto \( Z \) ; denote this linear map by \( f : {P}_{p + 1, * } \rightarrow Z \) .\n\nIn particular \( f \in {\operatorname{Hom}}_{A}^{p + 1, * }\left( {P, Z}\right) \) and \( {\delta f} = f \circ d = {d}^{2} = 0 \) . Since \( {\operatorname{Ext}}_{A}^{p + 1}\left( {M, - }\right) = \) 0 by hypothesis, \( f \) must satisfy \( f = g \circ d \) for some \( g : {P}_{p, * } \rightarrow Z \) . In other words, for \( x \in {P}_{p + 1, * },{dx} = f\left( x\right) = g\left( {dx}\right) \) . Thus \( g \) restricts to the identity in \( Z \) and \( {P}_{p, * } = Z \oplus \ker g \) . In particular \( \ker g \) is projective and\n\n\[M \leftarrow {P}_{0, * } \leftarrow \cdots \leftarrow {P}_{p - 1, * } \leftarrow \ker g \leftarrow 0\]\n\nis a projective resolution, whence \( \operatorname{proj}{\dim }_{A}\left( M\right) \leq p\left( M\right) \) .
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Yes
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Proposition 20.10 For any A-module, \( M \) , \[ {\operatorname{projdim}}_{A}\left( M\right) = \sup \left\{ {k \mid {\operatorname{Ext}}_{A}^{k}\left( {M, - }\right) \text{ is non-zero }}\right\} . \]
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proof: Denote the right hand side of the equation by \( p\left( M\right) \) . If \( M \) has a projective resolution of the form above then we can use it to compute \( {\operatorname{Ext}}_{A}^{n}\left( {M, - }\right) \) and it follows at once that \( {\operatorname{Ext}}_{A}^{n}\left( {M, - }\right) = 0, n > k \) . Hence \( p\left( M\right) \leq \operatorname{proj}{\dim }_{A}\left( M\right) \) . Conversely, suppose \( p\left( M\right) \) is finite and let \( \varrho : \left( {\left\{ {P}_{k, * }\right\}, d}\right) \rightarrow \left( {M,0}\right) \) be any projective resolution. For simplicity write \( p = p\left( M\right) \) . Let \( Z \subset {P}_{p, * } \) be the submodule of cycles \( \left( {Z = \ker d}\right) \) . Then the differential, \( d \), maps \( {P}_{p + 1, * } \) onto \( Z \) ; denote this linear map by \( f : {P}_{p + 1, * } \rightarrow Z \) . In particular \( f \in {\operatorname{Hom}}_{A}^{p + 1, * }\left( {P, Z}\right) \) and \( {\delta f} = f \circ d = {d}^{2} = 0 \) . Since \( {\operatorname{Ext}}_{A}^{p + 1}\left( {M, - }\right) = \) 0 by hypothesis, \( f \) must satisfy \( f = g \circ d \) for some \( g : {P}_{p, * } \rightarrow Z \) . In other words, for \( x \in {P}_{p + 1, * },{dx} = f\left( x\right) = g\left( {dx}\right) \) . Thus \( g \) restricts to the identity in \( Z \) and \( {P}_{p, * } = Z \oplus \ker g \) . In particular \( \ker g \) is projective and \[ M \leftarrow {P}_{0, * } \leftarrow \cdots \leftarrow {P}_{p - 1, * } \leftarrow \ker g \leftarrow 0 \] is a projective resolution, whence \( \operatorname{proj}{\dim }_{A}\left( M\right) \leq p\left( M\right) \) .
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Yes
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Example 1 The Eilenberg-Moore spectral sequence.
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Suppose \( \left( {M, d}\right) \) is a right \( \left( {A,{d}_{A}}\right) \) -module with a semifree resolution \( \left( {P, d}\right) \overset{ \simeq }{ \rightarrow } \) \( \left( {M, d}\right) \) as constructed in Proposition 20.11. If \( \left( {N, d}\right) \) is any left \( \left( {A,{d}_{A}}\right) \) -module then the filtration \( \{ P\left( k\right) \} \) defines the filtration\n\n\[ P\left( 0\right) { \otimes }_{A}N \subset P\left( 1\right) { \otimes }_{A}N \subset \cdots \subset P\left( k\right) { \otimes }_{A}N \subset \cdots \]\n\nof \( P{ \otimes }_{A}N \) .\n\nThis leads to the (fundamental) Eilenberg-Moore spectral sequence, whose \( {E}^{1} - \) term is just \( \left( {\{ V\left( k\right) \otimes H\left( A\right) \} ,\partial }\right) { \otimes }_{H\left( A\right) }H\left( N\right) \) . The left hand tensorand is an\n\n\( H\left( A\right) \) -free resolution of \( H\left( M\right) \) . Thus the \( {E}^{2} \) -term of the Eilenberg-Moore spectral sequence is given by\n\n\[ {E}_{k, * }^{2} = {\operatorname{Tor}}_{k}^{H\left( A\right) }\left( {H\left( M\right), H\left( N\right) }\right) .\n\nUnder simple hypotheses the spectral sequence converges to \( H\left( {P{ \otimes }_{A}N}\right) \) . As shown in \( §6 \), this homology is a functor, independent of the choice of semifree resolution. It is called the differential tor, Diff \( {\operatorname{Tor}}^{A}\left( {M, N}\right) \) and so the Eilenberg-Moore spectral sequence converges (usually) from\n\n\[ {\operatorname{Tor}}^{H\left( A\right) }\left( {H\left( M\right), H\left( N\right) }\right) \Rightarrow \text{ Diff }{\operatorname{Tor}}^{A}\left( {M, N}\right) . \]
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Yes
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If \( L \) is a graded Lie algebra then a representation ad : \( L \rightarrow \operatorname{Hom}\left( {L, L}\right) \) is defined by\n\n\[ \left( {\operatorname{ad}x}\right) \left( y\right) = \left\lbrack {x, y}\right\rbrack ,\;x, y \in L. \]
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This is called the adjoint representation, and (by the Jacobi identity) makes \( L \) into an \( L \) -module. The submodules are precisely the ideals in \( L \) .
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No
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The product of two graded Lie algebras \( E \) and \( L \) is the direct sum, \( E \oplus L \) , with Lie bracket\n\n\[ \left\lbrack {\left( {x, y}\right) ,\left( {{x}^{\prime },{y}^{\prime }}\right) }\right\rbrack = \left( {\left\lbrack {x,{x}^{\prime }}\right\rbrack ,\left\lbrack {y,{y}^{\prime }}\right\rbrack }\right) ,\;\begin{array}{l} x,{x}^{\prime } \in E \\ y,{y}^{\prime } \in L. \end{array} \]
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In particular for \( x \in E, y \in L \) we have \( \left\lbrack {x, y}\right\rbrack = 0 \) in \( E \oplus L \) .
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Yes
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Theorem 21.1 (Poincaré-Birkoff-Witt) Let \( L \) be a graded Lie algebra. Then\n\n(i) The admissible \( U \) -monomials are a basis of \( {UL} \) .\n\n(ii) In particular, the linear map \( \iota : L \rightarrow {UL} \) is an inclusion and extends to an isomorphism of graded vector spaces \( {\Lambda L}\overset{ \cong }{ \rightarrow }{UL} \) .
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proof: [139] If \( M = {\alpha }_{1},\ldots ,{\alpha }_{k} \) is admissible then so is \( N = {\alpha }_{2},\ldots ,{\alpha }_{k} \) and we write \( M = {\alpha }_{1}N \) . We show first that an \( L \) -module structure in \( {\Lambda L} \) is defined by the conditions \( {x}_{\alpha } \cdot {x}_{\phi } = {x}_{\alpha } \) and, if \( M = {\alpha }_{1}N \) ,\n\n\[ \n{x}_{\alpha } \cdot {x}_{M} = \left\{ \begin{array}{l} {x}_{\alpha M}\text{ if }\alpha < {\alpha }_{1}\text{ or }\alpha = {\alpha }_{1}\text{ and }\deg {x}_{\alpha }\text{ is even. } \\ \frac{1}{2}\left\lbrack {{x}_{\alpha },{x}_{\alpha }}\right\rbrack \cdot {x}_{N}\text{ if }\alpha = {\alpha }_{1}\text{ and }\deg {x}_{\alpha }\text{ is odd. } \\ \left\lbrack {{x}_{\alpha },{x}_{{\alpha }_{1}}}\right\rbrack \cdot {x}_{N} + {\left( -1\right) }^{\deg {x}_{\alpha }\deg {x}_{{\alpha }_{1}}}{x}_{{\alpha }_{1}} \cdot {x}_{\alpha } \cdot {x}_{N}\text{ if }\alpha > {\alpha }_{1}. \end{array}\right. \n\] \n\nIndeed suppose by (transfinite) induction that \( {x}_{{\alpha }^{\prime }} \cdot {x}_{{M}^{\prime }} \) is defined if length \( {M}^{\prime } < \) length \( M \) or if length \( {M}^{\prime } = \) length \( M \) and \( {\alpha }^{\prime } < \alpha \) . Assume further that \( {x}_{{\alpha }^{\prime }} \cdot {x}_{{M}^{\prime }} \) is a linear combination of monomials of length \( \leq \operatorname{length}{M}^{\prime } + 1 \) . Then the right hand side of the formula above is defined, and is a linear combination of monomials of length \( \leq \) length \( M + 1 \) . By induction \( {x}_{\alpha } \cdot {x}_{M} \) is defined for all \( \alpha \) and \( M \) .\n\nTo show that this makes \( {\Lambda L} \) into an \( L \) -module we write\n\n\[ \n\left( {\alpha ,\beta, M}\right) = \left\lbrack {{x}_{\alpha },{x}_{\beta }}\right\rbrack \cdot {x}_{M} - {x}_{\alpha } \cdot {x}_{\beta } \cdot {x}_{M} + {\left( -1\right) }^{\deg {x}_{\alpha }\deg {x}_{\beta }}{x}_{\beta } \cdot {x}_{\alpha } \cdot {x}_{M} \n\] \n\nand then verify that \( \left( {\alpha ,\beta, M}\right) = 0 \) for all \( \alpha ,\beta \) and \( M \) . For this we may clearly suppose \( \beta \leq \alpha \) and induct, as before, on length \( M \) and on \( \\b
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No
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Proposition 21.2 If \( L \) is any graded Lie algebra then a natural linear isomorphism of graded vector spaces, \[ \gamma : {\Lambda L}\overset{ \cong }{ \rightarrow }{UL} \] is given by \( \gamma \left( {{x}_{1} \land \cdots \land {x}_{k}}\right) = \frac{1}{k!}\mathop{\sum }\limits_{{\sigma \in {S}_{k}}}{\varepsilon }_{\sigma }{x}_{\sigma \left( 1\right) }\cdots {x}_{\sigma \left( k\right) } \) .
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proof: We adopt the notation of Theorem 21.1 and its proof. Let \( {UL}\left( k\right) \) denote the subspace spanned by monomials in \( L \) of length \( \leq k \) . The last part of the proof of 21.1 establishes in fact that the admissible monomials of length \( \leq k \) are a basis of \( {UL}\left( k\right) \) . Moreover, if \( {x}_{{\alpha }_{1}}\cdots {x}_{{\alpha }_{k}} \) is an admissible monomial in \( {UL}\left( k\right) \) and if \( \sigma \) is any permutation then \( {x}_{{\alpha }_{1}}\cdots {x}_{{\alpha }_{k}} - {\varepsilon }_{\sigma }{x}_{{\alpha }_{\sigma \left( 1\right) }}\cdots {x}_{{\alpha }_{\sigma \left( k\right) }} \in {UL}\left( {k - 1}\right) \) . Hence for any admissible sequence \( M \) of length \( k,\gamma \left( {x}_{M}\right) - {u}_{M} \in {UL}\left( {k - 1}\right) \) . Since the \( {u}_{M} \) with \( M \) of length \( k \) represent a basis of \( {UL}\left( k\right) /{UL}\left( {k - 1}\right) \) while the \( {x}_{M} \) are a basis of \( {\Lambda }^{k}L \), it follows that \( \gamma \) is an isomorphism.
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Yes
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Proposition 21.3 The inclusion \( L \rightarrow {UL} \) is an isomorphism of \( L \) onto the graded Lie algebra of primitive elements in UL.
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proof: It is immediate from the definition of \( {UL} \) that the inclusion is a morphism of Lie algebras. To see that it is an isomorphism onto \( {P}_{ * }\left( {UL}\right) \) define a Hopf algebra structure in \( {\Lambda L} \) with diagonal \( {\Delta }_{\Lambda } \) the unique algebra morphism given by \( {\Delta }_{\Lambda }\left( x\right) = x \otimes 1 + 1 \otimes x, x \in L \) . Since \( \Delta \) is an algebra morphism a very short computation shows that the linear isomorphism \( \gamma : {\Lambda L}\overset{ \cong }{ \rightarrow }{UL} \) of Proposition 21.2 is an isomorphism of coalgebras. Thus it is sufficient to prove that \( L \) is the primitive subspace of \( {\Lambda L} \) . Fix a basis \( \left\{ {x}_{\alpha }\right\} \) of \( L \) . Since \( {\Delta }_{\Lambda } \) is an algebra morphism, \( {\Delta }_{\Lambda }\left( {{x}_{{\alpha }_{1}}^{{k}_{1}} \land \cdots \land {x}_{{\alpha }_{r}}^{{k}_{r}}}\right) \) can be expanded easily. In particular it contains a term of the form \( {k}_{1}{x}_{\alpha }^{1} \otimes \left( {{x}_{{\alpha }_{1}}^{{k}_{1} - 1} \land \cdots \land {x}_{{\alpha }_{r}}^{{k}_{r}}}\right) \) and this term cannot appear in the diagonal of any other monomial in the \( \left\{ {x}_{\alpha }\right\} \) . Thus a linear combination of monomials is primitive if and only if they all have length 1 ; i.e., all primitive elements in \( {\Lambda L} \) are in \( L \) .
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Yes
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Theorem 21.6 The linear map\n\n\\[ \n\\sigma : {L}_{X} \\rightarrow L \n\\]\n\ndefined by \\( \\theta \\left( {s\\alpha }\\right) = {s\\sigma \\alpha },\\alpha \\in {L}_{X} \\), is an isomorphism of graded Lie algebras.
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proof: Theorem 15.11 implies that \\( \\theta \\) is an isomorphism of graded vector spaces. Hence so is \\( \\sigma \\) . To check that \\( \\sigma \\) preserves Lie brackets, recall that if \\( \\alpha ,\\beta \\in {L}_{X} \\) then their Lie bracket, as defined in \\( §{21}\\left( \\mathrm{\\;d}\\right) \\), satisfies\n\n\\[ \ns\\left\\lbrack {\\alpha ,\\beta }\\right\\rbrack = - {\\left( -1\\right) }^{\\deg \\alpha + \\deg \\beta }{\\partial }_{ * }^{-1}\\left\\lbrack {\\alpha ,\\beta }\\right\\rbrack \n\\]\n\n\\[ \n= \\;{\\left( -1\\right) }^{\\deg \\beta }{\\left\\lbrack {\\partial }_{ * }^{-1}\\alpha ,{\\partial }_{ * }^{-1}\\beta \\right\\rbrack }_{W} = {\\left( -1\\right) }^{\\deg \\alpha }{\\left\\lbrack s\\alpha, s\\beta \\right\\rbrack }_{W}\\text{ . } \n\\]\n\nOn the other hand Proposition 13.16 states that\n\n\\[ \n\\left\\langle {v;{\\left\\lbrack s\\alpha, s\\beta \\right\\rbrack }_{W}}\\right\\rangle = {\\left( -1\\right) }^{\\deg \\alpha + \\deg \\beta + 1}\\left\\langle {{d}_{1}v;{s\\alpha },{s\\beta }}\\right\\rangle ,\\;v \\in V. \n\\]\n\nThus the Lie bracket in \\( {L}_{X} \\) satisfies\n\n\\[ \n\\left\\langle {{d}_{1}v;{s\\alpha },{s\\beta }}\\right\\rangle = {\\left( -1\\right) }^{\\deg \\beta + 1}\\langle v;s\\left\\lbrack {\\alpha ,\\beta }\\right\\rbrack \\rangle \n\\]\n\nwhich is the defining condition for the Lie bracket in \\( L \\) .
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Yes
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Lemma 22.1 Suppose \( C = \mathbb{k} \oplus \bar{C} \) is a primitively cogenerated cocommutative graded coalgebra. Then any linear map of degree zero, \( f : \bar{C} \rightarrow V \) lifts to a unique morphism of graded coalgebras, \( \varphi : C \rightarrow {\Lambda V} \) such that \( {\left. {\xi }_{\varphi }\right| }_{\bar{C}} = f \) .
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proof: Define \( {f}^{\left( k\right) } : \bar{C} \otimes \cdots \otimes \bar{C} \rightarrow {\Lambda }^{k}V \) by\n\n\[ \n{f}^{\left( k\right) }\left( {{c}_{1} \otimes \cdots \otimes {c}_{k}}\right) = \frac{1}{k!}f\left( {c}_{1}\right) \land \cdots \land f\left( {c}_{k}\right) .\n\]\n\nRecall that \( {\bar{\Delta }}^{\left( 0\right) } = i{d}_{\bar{C}} \) and define \( \varphi \) by \( \varphi \left( 1\right) = 1 \) and\n\n\[ \n{\varphi c} = \mathop{\sum }\limits_{{k = 0}}^{\infty }{f}^{\left( k + 1\right) }{\bar{\Delta }}^{\left( k\right) }c,\;c \in \bar{C}.\n\]\n\n(Since \( C \) is primitively cogenerated, this is a finite sum.)\n\nTo verify that \( \varphi \) is a coalgebra morphism, write \( {\bar{\Delta }}^{\left( k - 1\right) }c = \mathop{\sum }\limits_{\alpha }{c}_{1}^{\alpha } \otimes \cdots \otimes \) \( {c}_{k}^{\alpha } \) . Co-commutativity implies that \( {\bar{\Delta }}^{\left( k - 1\right) }c = \mathop{\sum }\limits_{\alpha } \pm {c}_{\sigma \left( 1\right) }^{\alpha } \otimes \cdots \otimes {c}_{\sigma \left( k\right) }^{\alpha } \), for each permutation, \( \sigma \) .\n\nCo-associativity implies that \( {\bar{\Delta }}^{\left( k\right) } = \left( {{\bar{\Delta }}^{\left( p\right) } \otimes {\bar{\Delta }}^{\left( q\right) }}\right) \circ \bar{\Delta } \) for all \( p, q \) such that \( p + q = k - 1 \) . Finally, for any \( {v}_{i} \in V \) ,\n\n\[ \n\Delta \left( {{v}_{1} \land \cdots \land {v}_{k}}\right) = \left( {{v}_{1} \otimes 1 + 1 \otimes {v}_{1}}\right) \land \cdots \land \left( {{v}_{k} \otimes 1 + 1 \otimes {v}_{k}}\right)\n\]\n\n\[ \n= \mathop{\sum }\limits_{{p = 0}}^{k}\frac{1}{p!}\frac{1}{\left( {k - p}\right) !}\mathop{\sum }\limits_{{\sigma \in {S}_{k}}} \pm {v}_{\sigma \left( 1\right) } \land \cdots \land {v}_{\sigma \left( p\right) } \otimes {v}_{\sigma \left( {p + 1}\right) } \land \cdots \land {v}_{\sigma \left( k\right) }.\n\]\n\nCombining these facts in a straightforward calculation gives \( \left( {\varphi \otimes \varphi }\right) \Delta = {\Delta \varphi } \).\n\nTo prove uniqueness observe that any morphism maps \( \ker {\widetilde{\Delta }}^{\left( 1\right) } \) into \( V \) . Hence if \( \psi \) is a second morphism then for \( c \in \ker {\bar{\Delta }}^{\left( 1\right) },{\varphi c} = {\xi \varphi c} = f\left( c\right) = {\xi \psi c} = {\psi c} \) . Suppose \( \varphi \) and \( \psi \) agree in \( \ker {\bar{\Delta }}^{\left( n\right) } \) and let \( c \in \ker {\bar{\Delta }}^{\left( n + 1\right) } \) . By co-associativity, \( \bar{\Delta }c \in \ker {\bar{\Delta }}^{\left( n\right) } \otimes \ker {\bar{\Delta }}^{\left( n\right) } \) . Thus \( \bar{\Delta }{\varphi c} = \left( {\varphi \otimes \varphi }\right) \bar{\Delta }c = \left( {\psi \otimes \psi }\right) \bar{\Delta }c = \bar{\Delta }{\psi c} \) . This implies that \( \left( {\varphi - \psi }\right) c = \left( {{\xi \varphi } - {\xi \psi }}\right) c = 0 \) .
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Yes
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Lemma 22.2 \( {\theta }_{g} \) is a coderivation in \( {\Lambda V} \) . It is the unique coderivation that extends \( g \) and decreases wordlength by \( k - 1 \) .
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proof: The coderivation property is a simple calculation. Uniqueness is proved in the same way as in Lemma 22.1.
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No
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Proposition 22.3 The inclusion \( \mathbf{k} \rightarrow \left( {{\Lambda sL} \otimes {UL}, d}\right) \) is a quasi-isomorphism.
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proof of 22.3: In the proof of Proposition 21.2 we introduced the subspaces \( {UL}\left( n\right) \subset {UL} \) spanned linearly by the monomials \( {x}_{1}\cdots {x}_{k},{x}_{i} \in L, k \leq n \) . We showed there that a linear isomorphism \( \gamma : {\Lambda L}\overset{ \cong }{ \rightarrow }{UL} \) is given by \( {x}_{1} \land \cdots \land {x}_{k} \mapsto \) \( \frac{1}{k!}\mathop{\sum }\limits_{\sigma }{\varepsilon }_{\sigma }{x}_{\sigma \left( 1\right) }\cdots {x}_{\sigma \left( k\right) } \), and that it induces isomorphisms \( {\Lambda }^{k}L\overset{ \cong }{ \rightarrow }{UL}\left( k\right) /{UL}(k - \) 1). Now write \( {\Lambda sL} \otimes {UL} \) as the increasing union of the graded subspaces \( {F}_{k} = \) \( \mathop{\sum }\limits_{{i + j \leq k}}{\Lambda }^{ \leq i}{sL} \otimes {UL}\left( j\right) \) . As usual, identify \( {\Lambda sL} \otimes {\Lambda L} = \Lambda \left( {{sL} \oplus L}\right) \) . Then \[ {id} \otimes \gamma : {\Lambda }^{ \leq k}\left( {{sL} \oplus L}\right) \overset{ \cong }{ \rightarrow }{F}_{k},\;k \geq 0, \] and so induces isomorphisms \( {\Lambda }^{k}\left( {{sL} \oplus L}\right) \overset{ \cong }{ \rightarrow }{F}_{k}/{F}_{k - 1} \) . Next observe that the differential, \( d \), preserves the spaces \( {F}_{k} \) . Moreover, because \( s\left\lbrack {{x}_{i},{x}_{j}}\right\rbrack \) has lower wordlength than \( s{x}_{i} \land s{x}_{j} \) the corresponding terms in the formula for \( d \) (see (b), above) disappear in \( {F}_{k}/{F}_{k - 1} \) . Also, \( x \cdot \gamma \left( {{y}_{1} \land \cdots \land }\right. \) \( \left. {y}_{k}\right) - \gamma \left( {x \land {y}_{1} \land \cdots \land {y}_{k}}\right) \in {UL}\left( k\right) \) . These comments imply that \( {id} \otimes \gamma \) is an isomorphism of complexes \[ \left( {{\Lambda }^{k}\left( {{sL} \oplus L}\right) ,\delta }\right) \overset{ \cong }{ \rightarrow }\left( {{F}_{k}/{F}_{k - 1},\bar{d}}\right) \] where \( \delta \) is the derivation in the algebra \( \Lambda \left( {{sL} \oplus L}\right) \) specified by \[ {\delta x} = {dx}\;\text{ and }\;{\delta sx} = {\left( -1\right) }^{\deg x + 1}x - {sdx},\;x \in L. \] Trivially, \( H\left( {{sL} \oplus L,\delta }\right) = 0 \) . Choose a basis of \( {sL} \oplus L \) of the form \( \left\{ {{u}_{\alpha },\delta {u}_{\alpha }}\right\} \) . Then \( \Lambda \left( {{sL} \oplus L,\delta }\right) \cong \bigotimes \Lambda \left( {{u}_{\alpha },\delta {u}_{\alpha }}\right) \) and so \( H\left( {\Lambda \left( {{sL} \oplus L}\right) ,\delta }\right) = \mathbb{k} \) ; i.e. \( H\left( {{\Lambda }^{k}\left( {{sL} \oplus L}\right) ,\delta }\right) = 0, k \geq 1 \) . Because of the isomorphism above, \( H\left( {{F}_{k}/{F}_{k - 1},\bar{d}}\right) \) \( = 0, k \geq 1 \) . It follows by induction on \( k \) that \( H\left( {{F}_{k}/{F}_{0},\bar{d}}\right) = 0, k \geq 1 \), and so \( H\left( {\left( {{\Lambda sL} \otimes {UL}}\right) /\mathbb{k}, d}\right) = 0 \), as desired.
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Yes
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Proposition 22.4 Right multiplication makes \( \\left( {{\\Lambda sL} \\otimes {UL}, d}\\right) \) into a right semifree \( \\left( {{UL}, d}\\right) \) -module.
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proof of 22.4: It is immediate from the definitions that multiplication from the right makes \( {C}_{ * }\\left( {L;{UL}}\\right) = {\\Lambda sL} \\otimes {UL} \) into a right \( \\left( {{UL}, d}\\right) \) -module. Moreover the subspaces \( M\\left( k\\right) = {\\Lambda }^{ \\leq k}{sL} \\otimes {UL} \) define an increasing family of submodules, and\n\n\[ \n\\left( {M\\left( k\\right) /M\\left( {k - 1}\\right), d}\\right) \\cong \\left( {{\\Lambda }^{k}{sL},{d}_{0}}\\right) \\otimes \\left( {{UL}, d}\\right) .\n\]\n\nThe two step filtration \( \\left( {\\ker {d}_{0}}\\right) \\otimes \\left( {{UL}, d}\\right) \\subset \\left( {{\\Lambda }^{k}{sL},{d}_{0}}\\right) \\otimes \\left( {{UL}, d}\\right) \) exhibits \( \\left( {M\\left( k\\right) /M\\left( {k - 1}\\right), d}\\right) \) as \( \\left( {{UL}, d}\\right) \) -semifree. Now Lemma 6.3 asserts that \( {C}_{ * }\\left( {L;{UL}}\\right) \) is \( \\left
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Yes
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Proposition 22.7 A natural dgc quasi-isomorphism \[ \lambda : {C}_{ * }\left( L\right) \overset{ \simeq }{ \rightarrow }{BUL} \] is given by \( \lambda : s{x}_{1} \land \cdots \land s{x}_{k} \mapsto \mathop{\sum }\limits_{{\sigma \in {S}_{k}}}{\varepsilon }_{\sigma }\left\lbrack {s{x}_{\sigma \left( 1\right) }\left| \cdots \right| s{x}_{\sigma \left( k\right) }}\right\rbrack \), where \( s{x}_{1} \land \cdots \land s{x}_{k} = \) \( {\varepsilon }_{\sigma }s{x}_{\sigma \left( 1\right) } \land \cdots \land s{x}_{\sigma \left( k\right) }.
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proof: A straightforward, if tedious, calculation verifies that \( \lambda \) is a morphism of dgc's. Similarly, \[ \lambda \otimes {id} : \left( {{\Lambda sL} \otimes {UL}, d}\right) \rightarrow \left( {{BUL} \otimes {UL}, d}\right) \] is a morphism of \( \left( {{UL}, d}\right) \) -modules. According to Proposition 19.2, \( H({BUL} \otimes \) \( {UL}) = \mathbb{R} \) and \( \left( {{BUL} \otimes {UL}, d}\right) \) is \( \left( {{UL}, d}\right) \) -semifree. According to Propositions 22.3 and 22.4, \( \left( {{\Lambda sL} \otimes {UL}, d}\right) \) has the same two properties. Because \( H\left( {{BUL} \otimes {UL}}\right) = \mathbb{R} = H\left( {{\Lambda sL} \otimes {UL}}\right) ,\lambda \otimes {id} \) is necessarily a quasi-isomorphism. Because both are \( \left( {{UL}, d}\right) \) -semifree, Proposition 6.7 (ii) asserts that \( \left( {\lambda \otimes {id}}\right) { \otimes }_{UL}\mathbb{k} \) is a quasi-isomorphism. But, trivially, \( \left( {\lambda \otimes {id}}\right) { \otimes }_{UL}\mathbb{k} = \lambda \).
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Yes
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Proposition 22.8 The linear map \( \varrho \) is a natural quasi-isomorphism of complexes,\n\n\[ \left( {{C}_{ * }\left( {\mathbb{L}}_{V}\right), d}\right) \overset{ \simeq }{ \rightarrow }\left( {{sV} \oplus \mathbb{k},\bar{d}}\right) \]
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proof: It is immediate from the definitions that \( \varrho \) commutes with the differentials. Moreover, since \( U{\mathbb{L}}_{V} = {TV} \), there is an analogous morphism \( {\varrho }^{\prime } \) : \( \left( {{BU}{\mathbb{L}}_{V}, d}\right) \rightarrow \left( {{sV} \oplus \mathbb{k},\bar{d}}\right) \) constructed in \( §{19} \) just before Proposition 19.1. Moreover, Proposition 19.1 asserts that \( {\varrho }^{\prime } \) is a quasi-isomorphism.\n\nIt is immediate that the quasi-isomorphism of Proposition 22.7, \( \lambda : {C}_{ * }\left( {\mathbb{L}}_{V}\right) \overset{ \simeq }{ \rightarrow } \) \( {BU}{\mathbb{L}}_{V} \), satisfies \( {\varrho }^{\prime }\lambda = \varrho \) . Thus \( \varrho \) is a quasi-isomorphism.
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Yes
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Proposition 22.12 If \( \varphi : \left( {{\mathbb{L}}_{W}, d}\right) \rightarrow \left( {{\mathbb{L}}_{V}, d}\right) \) is a morphism of free connected chain Lie algebras then\n\n\( \varphi \) is a quasi-isomorphism \( \Leftrightarrow {\varphi }_{0} \) is a quasi-isomorphism.
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proof: According to Proposition 22.5 and Proposition 22.8 respectively,\n\n\[ \varphi \text{is a quasi-isomorphism} \Leftrightarrow {C}_{ * }\left( \varphi \right) \text{is a quasi-isomorphism} \]\n\n\( \Leftrightarrow {\varphi }_{0} \) is a quasi-isomorphism.
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Yes
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Lemma 23.1 If \( \left( {L, d}\right) \) is a connected chain Lie algebra and each \( {L}_{i} \) is finite dimensional then\n\n\[ \sigma : \Lambda {\left( sL\right) }^{\sharp }\overset{ \cong }{ \rightarrow }{C}^{ * }\left( L\right) \]\n\n is an isomorphism of graded algebras, which exhibits \( {C}^{ * }\left( L\right) \) as a Sullivan algebra.
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proof: Let \( {y}_{i} = s{x}_{i} \) be a basis for \( {sL} \) and let \( {v}_{j} \) be the dual basis for \( {\left( sL\right) }^{\sharp } \) : \( \left\langle {{v}_{j},{y}_{i}}\right\rangle = {\delta }_{ij} \) . If \( v \in {\left( sL\right) }^{\sharp } \) and \( \Phi \in {\Lambda }^{p}{\left( sL\right) }^{\sharp } \) then\n\n\[ \left\langle {v \land \Phi ,{y}_{{i}_{1}} \land \cdots \land {y}_{{i}_{p + 1}}}\right\rangle = \left\langle {v \otimes \Phi ,\Delta \left( {{y}_{{i}_{1}} \land \cdots \land {y}_{{i}_{p + 1}}}\right) }\right\rangle ,\;\text{and} \]\n\n\[ C = \mathop{\sum }\limits_{{j = 1}}^{{p + 1}}{\left( -1\right) }^{\deg {y}_{{i}_{j}}\deg \Phi }\left\langle {v,{y}_{{i}_{j}}}\right\rangle \left\langle {\Phi ,{y}_{{i}_{1}} \land \cdots {\widehat{y}}_{{i}_{j}}\cdots \land {y}_{{i}_{p + 1}}}\right\rangle . \]\n\nIt follows that\n\n\[ \left\langle {{v}_{1}^{{k}_{1}} \land \cdots \land {v}_{n}^{{k}_{n}},{y}_{n}^{{k}_{n}} \land \cdots \land {y}_{1}^{{k}_{1}}}\right\rangle = {k}_{1}!\cdots {k}_{n}! \]\n\n(where \( {k}_{i} \leq 1 \) if \( \left| {v}_{i}\right| \) is odd) and that \( {v}_{1}^{{k}_{1}} \land \cdots \land {v}_{n}^{{k}_{n}} \) evaluates all other monomials to zero. Since char \( \mathbb{k} = 0 \) this implies that \( \sigma \) is an isomorphism, because \( {C}_{ * }\left( L\right) \) is finite dimensional in each degree.\n\nNow \( \sigma \) identifies \( {C}^{ * }\left( L\right) \) as a cochain algebra of the form \( \left( {{\Lambda V}, d}\right) \) with \( V = \) \( {\left\{ {V}^{p}\right\} }_{p > 2} \) . Such a cochain algebra is always a Sullivan algebra. Indeed define \( V\left( 0\right) \subset V\left( 1\right) \subset \cdots \) by \( V\left( 0\right) = V \cap \ker d \) and \( V\left( k\right) = V \cap {d}^{-1}\left( {{\Lambda V}\left( {k - 1}\right) }\right) \) . Since \( {\left( \Lambda V\right) }^{3} = {V}^{3}, d\left( {V}^{2}\right) \subset {V}^{3} \) and \( {V}^{2} \subset V\left( 1\right) \subset \mathop{\bigcup }\limits_{k}V\left( k\right) \) . Suppose \( {V}^{ \leq n - 1} \subset \mathop{\bigcup }\limits_{k}V\left( k\right) \) . For \( v \in {V}^{n},{dv} = w + \Phi \) with \( w \in {V}^{n + 1} \) and \( \Phi \in \Lambda {V}^{ \leq n - 1} \) . Thus \( {dw} = - {d\Phi } \in \) \( \Lambda \left( {\mathop{\bigcup }\limits_{k}V\left( k\right) }\right) \) and so \( w \in \mathop{\bigcup }\limits_{k}V\left( k\right) \) . But then also \( v \in \mathop{\bigcup }\limits_{k}V\left( k\right) \) and it follows that \( V = \mathop{\bigcup }\limits_{k}V\left( k\right) \)
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Yes
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Proposition 23.2 Suppose \( \left( {L,{d}_{L}}\right) \) is a connected chain Lie algebra of finite type and each \( {L}_{i} \) is finite dimensional. Then\n\n(i) \( {C}^{ * }\left( {L,{d}_{L}}\right) = \left( {{\Lambda V}, d}\right) \) and \( V \) and \( {sL} \) are dual graded vector spaces.\n\n(ii) \( d = {d}_{0} + {d}_{1} \) is the sum of its linear and quadratic parts (§12(a) and \( §{13}\left( e\right) \) ), and\n\n\[ \left\langle {{d}_{0}v;{sx}}\right\rangle = {\left( -1\right) }^{\deg v}\left\langle {v;s{d}_{L}x}\right\rangle \text{ and }\left\langle {{d}_{1}v;{sx} \land {sy}}\right\rangle = {\left( -1\right) }^{\deg y + 1}\langle v;s\left\lbrack {x, y}\right\rbrack \rangle . \]
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proof: The first assertion follows from Lemma 23.1 and the definition of the differential in \( {C}_{ * }\left( {L,{d}_{L}}\right) \) . For the second assertion let \( L \) be the desuspension of \( \operatorname{Hom}\left( {V,\mathbb{R}}\right) \) and use the formulae above to define \( {d}_{L} : L \rightarrow L \) and \( \left\lbrack ,\right\rbrack : L \times L \rightarrow \) \( L \) . The equation \( {d}^{2} = 0 \) reduces to \( {d}_{1}^{2} = 0,{d}_{0}{d}_{1} + {d}_{1}{d}_{0} = 0 \) and \( {d}_{0}^{2} = 0 \) . These translate respectively to: \( \left\lbrack \right\rbrack \) is a Lie bracket \( \left( {§{21}\left( \mathrm{e}\right) }\right) ,{d}_{L} \) is a Lie derivation and \( {d}_{L}^{2} = 0 \) .
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Yes
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Suppose \( L = {\left\{ {L}_{i}\right\} }_{i \geq 1} \) is a graded Lie algebra and each \( {L}_{i} \) is finite dimensional. Then Proposition 23.2 reduces to\n\n\[ \n{C}^{ * }\left( {L,0}\right) = \left( {{\Lambda V},{d}_{1}}\right) \n\]\n\nin which \( {sL} \) and \( V \) are dual graded vector spaces and \( {d}_{1} \) is purely quadratic. Conversely, any commutative cochain algebra of the form \( \left( {{\Lambda V},{d}_{1}}\right) \) with \( V = \) \( {\left\{ {V}^{p}\right\} }_{p > 2} \), each \( {V}^{p} \) finite dimensional and \( {d}_{1} \) is purely quadratic determines a graded Lie algebra \( L \) by the requirement that \( \left( {{\Lambda V},{d}_{1}}\right) = {C}^{ * }\left( {L,0}\right) \) .
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Next notice that Propositions 22.3 and 22.4 identify \( {C}_{ * }\left( {L;{UL}}\right) \) as an exact sequence\n\n\[ \n0 \leftarrow \mathbb{R}\overset{\varepsilon }{ \leftarrow }{UL}\overset{d}{ \leftarrow }{sL} \otimes {UL}\overset{d}{ \leftarrow }{\Lambda }^{2}{sL} \otimes {UL}\overset{d}{ \leftarrow }\cdots \n\]\n\nof right \( {UL} \) -modules. In other words (cf. \( §{20}\left( \mathrm{a}\right) \) ) this is a free \( {UL} \) -resolution of the trivial \( {UL} \) -module, \( \mathbb{R} \) . On the other hand, it is immediate from the definitions that \( {C}^{ * }\left( L\right) = {\operatorname{Hom}}_{UL}\left( {{C}_{ * }\left( {L;{UL}}\right) ,\mathbb{k}}\right) \), and so\n\n\[ \nH\left( {{C}^{ * }\left( L\right) }\right) = {\operatorname{Ext}}_{UL}\left( {\mathbb{I}k,\mathbb{R}}\right) . \n\]
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No
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Example 2 Free graded Lie algebras.\n\nSuppose \( E = {\left\{ {E}^{i}\right\} }_{i \geq 2} \) is a graded vector space of finite type and \( \left( {H,0}\right) \) is the commutative cochain algebra with zero differential defined by\n\n\[ H = \mathbb{I}k \oplus E \\text{ and } E \\cdot E = 0.\]\n\nThe dual graded coalgebra has the form \( C = \\mathbb{R} \\oplus \\bar{C} \) with \( \\bar{\\Delta } = 0 : \\bar{C} \\rightarrow \\bar{C} \\otimes \\bar{C} \) . Thus the differential in \( {\\mathcal{L}}_{\\left( C,0\\right) } \) is zero \( \\left( {§{22}\\left( \\mathrm{e}\\right) }\\right) \) . In other words, if \( W \) is the graded vector space defined by \( {W}_{i} = \\operatorname{Hom}\\left( {{E}^{i + 1},\\mathbb{k}}\\right) \) then\n\n\[ {\\mathcal{L}}_{\\left( H,0\\right) } = {\\mathcal{L}}_{\\left( C,0\\right) } = \\left( {{\\mathbb{L}}_{W},0}\\right) .\]
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Thus in this case dualizing Theorem 22.9 provides a cochain algebra quasi-isomorphism\n\n\[ {C}^{ * }\\left( {{\\mathbb{L}}_{W},0}\\right) \\overset{ \\simeq }{ \\rightarrow }\\left( {H,0}\\right) \]\n\nand \( {C}^{ * }\\left( {{\\mathbb{L}}_{W},0}\\right) \) is a minimal Sullivan algebra with purely quadratic differential. In other words, this is the minimal Sullivan model of \( \\left( {H,0}\\right) \), and so we recover the construction of Example 7, \( §{12}\\left( \\mathrm{\\;d}\\right) \) .
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Yes
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Example 4 Sullivan algebras \( \left( {{\Lambda W}, d}\right) \) for which \( {H}^{2k}\left( {{\Lambda W}, d}\right) = 0,1 \leq k \leq n \) .
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Here we consider minimal Sullivan algebras \( \left( {{\Lambda W}, d}\right) \) such that \( W = {\left\{ {W}^{i}\right\} }_{i > 2} \) is a graded vector space of finite type. The surjection \( \left( {{\Lambda }^{ + }W, d}\right) \rightarrow \left( {W,0}\right) \) with kernel \( {\Lambda }^{ \geq 2}W \) induces a linear map \( \zeta = \left\{ {{\zeta }^{p} : {H}^{p}\left( {\Lambda W}\right) \rightarrow {W}^{p}}\right\} \) . We shall use Lie models to show that\n\n\[ \n{H}^{2k}\left( {{\Lambda W}, d}\right) = 0,1 \leq k \leq n \Rightarrow {\zeta }^{p}\text{is injective,}\;p \leq {2n} + 1.\n\]\n\nIn fact, let \( \left( {{\mathbb{L}}_{V},\partial }\right) \) be a minimal Lie model for \( {\mathcal{L}}_{\left( \Lambda W, d\right) } \) (Theorem 22.13). Then (Example 3), \( {H}^{ + }\left( {{\Lambda W}, d}\right) \cong \operatorname{Hom}\left( {{sV},\mathbb{R}}\right) \) as graded vector spaces. Thus \( {V}_{{2k} - 1} = 0,1 \leq k \leq n \) and \( {\mathbb{L}}_{{V}_{ \leq {2n}}} \) is concentrated in even degrees. For degree reasons the differential in \( {\mathbb{L}}_{{V}_{ < {2n}}} \) is then zero.\n\nRecall from Example 3 that \( \eta : \left( {{\mathbb{L}}_{V},\partial }\right) \rightarrow \left( {V,0}\right) \) is the surjection with kernel \( \left\lbrack {{\mathbb{L}}_{V},{\mathbb{L}}_{V}}\right\rbrack \) . Since \( \partial \) vanishes in degrees \( \leq {2n},{H}_{i}\left( \eta \right) \) is surjective for \( i \leq {2n} \) . It follows that \( \operatorname{Hom}\left( {{sH}\left( \eta \right) ,\mathbb{R}}\right) \) is injective in degrees \( p \leq {2n} + 1 \) . Example 3 identifies this dual with \( \zeta \) ; i.e. \( {\zeta }^{p} \) is injective, \( p \leq {2n} + 1 \) .
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Yes
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Proposition 23.3 With the hypotheses above, \( Q\left( m\right) \) induces an isomorphism \( W\overset{ \cong }{ \rightarrow }H\left( {V,{d}_{0}}\right) \) . Its desuspended dual is an isomorphism of graded Lie algebras,\n\n\[ H\left( L\right) \overset{ \cong }{ \rightarrow }E \]
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proof: Use Theorem 14.9 and Theorem 14.11 to extend \( m \) to an isomorphism of the form\n\n\[ \left( {{\Lambda W}, D}\right) \otimes \left( {\Lambda \left( {U \oplus {\delta U}}\right) ,\delta }\right) \overset{ \cong }{ \rightarrow }\left( {{\Lambda V},{d}_{0} + {d}_{1}}\right) \]\n\nwith \( \delta : U\overset{ \cong }{ \rightarrow }{\delta U} \) . It follows that \( Q\left( m\right) \) induces an isomorphism\n\n\[ {\Lambda H}\left( {Q\left( m\right) }\right) : \left( {{\Lambda W},{D}_{1}}\right) \overset{ \cong }{ \rightarrow }\left( {{\Lambda H}\left( {V,{d}_{0}}\right), H\left( {d}_{1}\right) }\right) . \]\n\nNow \( V \) is dual to \( {sL} \) and \( {d}_{0} \) is dual (up to sign and suspension) to \( {d}_{L} \) . This identifies \( H\left( {V,{d}_{0}}\right) \) as the dual of \( {sH}\left( L\right) \) and \( \left( {{\Lambda H}\left( {V,{d}_{0}}\right), H\left( {d}_{1}\right) }\right) \) as \( {C}^{ * }\left( {H\left( L\right) ,0}\right) \) . Since \( \left( {{\Lambda W},{D}_{1}}\right) = {C}^{ * }\left( {E,0}\right) \), the ’desuspended’ dual of \( H\left( {Q\left( m\right) }\right) \) is an isomorphism from \( H\left( L\right) \) to the homotopy Lie algebra \( E \) .
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Yes
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Let \( L \) be a connected chain Lie algebra with each \( {L}_{i} \) finite dimensional. Then the homotopy Lie algebra of \( {C}^{ * }\left( {L,{d}_{L}}\right) \) is just \( H\left( L\right) \) (Proposition 23.3) and so the Milnor-Moore spectral sequence for \( {C}^{ * }\left( {L,{d}_{L}}\right) \) converges from
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\[ {E}_{2} = {\operatorname{Ext}}_{{UH}\left( L\right) }\left( {\mathbb{k},\mathbb{k}}\right) \Rightarrow H{C}^{ * }\left( {L,{d}_{L}}\right) . \]
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Yes
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Suppose \( \\left( {{\\Lambda W}, D}\\right) \) is the minimal Sullivan model for a simply connected topological space \( X \) with rational homology of finite type. Then \( H\\left( {{\\Lambda W}, D}\\right) \\cong \) \( {H}^{ * }\\left( {X;\\mathbb{R}}\\right) ,{H}_{ * }\\left( {{\\Omega X};\\mathbb{R}}\\right) \\cong U{L}_{X} \) and \( {L}_{X} \) is isomorphic to the homotopy Lie algebra of \( \\left( {{\\Lambda W}, D}\\right) \) .
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(This follows, respectively, from Corollary 10.10, Theorem 21.5 and Proposition 21.6.)
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No
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The free Lie model of a sphere.
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In the tensor algebra \( T\left( v\right) \) on a single generator, \( v \), the free Lie subalgebra \( \mathbb{L}\left( v\right) \) is given by\n\n\[ \mathbb{L}\left( v\right) = \left\{ \begin{array}{ll} \mathbf{k}v & \text{ if }\deg v = {2n} \\ \mathbf{k}v \oplus \mathbf{k}\left\lbrack {v, v}\right\rbrack & \text{ if }\deg v = {2n} + 1. \end{array}\right. \]\n\nThus the cochain algebra \( {C}^{ * }\left( {\mathbb{L}\left( v\right) }\right) \) is given, respectively, by\n\n\[ {C}^{ * }\left( {\mathbb{L}\left( v\right) }\right) = \left\{ \begin{array}{ll} \left( {\Lambda \left( e\right) ,0}\right) & ,\deg e = {2n} + 1 \\ \left( {\Lambda \left( {e,{e}^{\prime }}\right), d{e}^{\prime } = {e}^{2}}\right) & ,\deg e = {2n} + 2. \end{array}\right. \]\n\nThe reader will recognize these as the minimal Sullivan models for spheres constructed in Example 1, \( §{12}\left( \mathrm{a}\right) \) . In other words, we have for all \( n \geq 1 \) a quasi-isomorphism\n\n\[ {C}^{ * }\left( {\mathbb{L}\left( v\right) }\right) \overset{ \simeq }{ \rightarrow }{A}_{PL}\left( {S}^{n + 1}\right) \;,\deg v = n \]\n\nwhich exhibits \( \left( {\mathbb{L}\left( v\right) ,0}\right) \) as a minimal free Lie model for \( {S}^{n + 1} \) .
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Yes
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Proposition 24.4 With the hypotheses above the diagram\n\n\n\ncommutes, and identifies \( {sH}\left( \eta \right) \) with the Hurewicz homomorphism hur \( x \) .
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proof: The quasi-isomorphism \( {C}_{ * }\left( {\mathbb{L}}_{V}\right) \rightarrow {sV} \oplus \mathbb{k} \) of Proposition 22.8 converts the inclusion \( s\left( {{\mathbb{L}}_{V}, d}\right) \rightarrow {C}_{ * }\left( {\mathbb{L}}_{V}\right) \) into \( {s\eta } \) . Thus we have to show that\n\n\n\ncommutes. But this is precisely the dual of the commutative diagram at the end of \( §{13}\left( \mathrm{c}\right) \) .
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Yes
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Theorem 24.5 The following conditions on a simply connected topological space \( X \) are equivalent\n\n(i) The rational Hurewicz homomorphism hur \( {}_{X} : {\pi }_{ * }\left( X\right) \otimes \mathbb{Q} \rightarrow {H}_{ + }\left( {X;\mathbb{Q}}\right) \) is surjective.\n\n(ii) There is a rational homotopy equivalence of the form \( \mathop{\bigvee }\limits_{{\alpha \in \mathcal{I}}}{S}^{{n}_{\alpha }} \rightarrow X \) (each\n\n\( \left. {{n}_{\alpha } \geq 2}\right) \) .\n\n(iii) There is a well based, path connected space \( Y \) and a rational homotopy equivalence of the form \( {\sum Y} \rightarrow X \) .\n\n(iv) The rational homotopy Lie algebra \( {L}_{X} \) is a free graded Lie algebra.
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proof of Theorem 24.5: If (i) holds then we can choose based continuous maps \( {f}_{\alpha } : {S}^{{n}_{\alpha }} \rightarrow X \) so that the homology classes \( {H}_{ * }\left( {f}_{\alpha }\right) \left\lbrack {S}^{{n}_{\alpha }}\right\rbrack \) are a basis of \( {H}_{ + }\left( {X;\mathbb{Q}}\right) \) . Thus the map \( f : \mathop{\bigvee }\limits_{\alpha }{S}^{{n}_{\alpha }} \rightarrow X \) defined by the \( {f}_{\alpha } \) induces an isomorphism of rational homology. Hence \( f \) is a rational homotopy equivalence (Theorem 8.6) and (i) \( \Rightarrow \) (ii).\n\nClearly (ii) \( \Rightarrow \) (iii). Suppose (iii) holds. Use the Cellular models theorem 1.4 to reduce to the case \( Y \) is a CW complex with a single 0 -cell. Moreover, since \( {L}_{\sum Y} \cong {L}_{X} \) we may take \( X = {\sum Y} \).\n\nNow for any finite subcomplex \( Z \subset Y \) we know from Proposition 13.9 that \( {\sum Z} \) has a commutative model of the form \( \mathbb{R} \oplus H \) with zero differential and \( H \cdot H = 0 \) . Hence \( {\mathcal{L}}_{{Ik} \oplus H} \) is a Lie model for \( {\sum Z} \) and clearly this has the form \( \left( {{\mathbb{L}}_{W},0}\right) \) . Thus \( {L}_{Z} \cong H\left( {\mathbb{L}}_{W}\right) = {\mathbb{L}}_{W}
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Yes
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Proposition 24.6 With the hypotheses and notation immediately above\n\n(i) \( \ker {hu}{r}_{X} = s\left\lbrack {{L}_{X},{L}_{X}}\right\rbrack \) .\n\n(ii) The inclusion of \( W \) extends to an isomorphism \( {\mathbb{L}}_{W}\overset{ \cong }{ \rightarrow }{L}_{X} \) .
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proof: (i) It is clearly sufficient to prove this for a finite wedge of spheres. But then \( {H}_{ * }\left( {X;\mathbb{Q}}\right) \) has finite type and \( X \) is a suspension. Thus (Proposition 13.9) \( X \) has a commutative model of the form \( \mathbb{R} \oplus H \) with zero differential and \( H \cdot H = 0 \) . Now \( {\mathcal{L}}_{{Ik} \oplus H} \) is a Lie model for \( X \) and clearly has the form \( {\mathcal{L}}_{{Ik} \oplus H} = \left( {{\mathbb{L}}_{V},0}\right) \) . In particular, \( {L}_{X} = H\left( {\mathbb{L}}_{V}\right) = {\mathbb{L}}_{V} \) . Moreover, Proposition 24.4 identifies \( {hu}{r}_{X} \) with the surjection \( {s\eta } : s{\mathbb{L}}_{V} \rightarrow {sV} \), whose kernel is precisely \( s\left\lbrack {{\mathbb{L}}_{V},{\mathbb{L}}_{V}}\right\rbrack \) .\n\n(ii) It follows from (i) that \( {L}_{X} = W \oplus \left\lbrack {{L}_{X},{L}_{X}}\right\rbrack \) . Since \( {L}_{X} \) is free\n\nProposition 21.4 states precisely that \( {\mathbb{L}}_{W}\overset{ \cong }{ \rightarrow }{L}_{X} \) .
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Yes
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Example 1 \( \left( {{\mathbb{L}}_{V},0}\right) \) is the minimal Lie model of a wedge of spheres.
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In fact Theorem 24.7 asserts that for any \( V = {\left\{ {V}_{i}\right\} }_{i > 1} \) of finite type with basis \( \left\{ {v}_{\alpha }\right\} ,\left( {{\mathbb{L}}_{V},0}\right) \) is a Lie model of the space \( X = \mathop{\bigvee }\limits_{\alpha }{S}^{{n}_{\alpha } + 1} = {pt}{ \cup }_{f}\left( {\mathop{\coprod }\limits_{\alpha }{D}^{{n}_{\alpha } + 1}}\right) \) , \( \deg {v}_{\alpha } = {n}_{\alpha } \) . Note that this also follows from \( §{24}\left( \mathrm{c}\right) \) .
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Yes
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The free product of Lie models is a Lie model for a wedge, \( \mathop{\bigvee }\limits_{\alpha }{X}_{\alpha } \) .
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Suppose \( {X}_{\alpha } \) are simply connected spaces and \( X = \mathop{\bigvee }\limits_{\alpha }{X}_{\alpha } \) has rational homology of finite type. Let \( \left( {{\mathbb{L}}_{V\left( \alpha \right) },{d}_{\alpha }}\right) \) be a Lie model for \( {X}_{\alpha } \). Recall the free product defined in \( §{21}\left( \mathrm{c}\right) \) and note that \( {\coprod }_{\alpha }\left( {{\mathbb{L}}_{V\left( \alpha \right) },{d}_{\alpha }}\right) \) is the free \( \operatorname{dgl}\left( {{\mathbb{L}}_{{\bigoplus }_{\alpha }V\left( \alpha \right) }, d}\right) \), in which \( d \) restricts to \( {d}_{\alpha } \) in each \( V\left( \alpha \right) \). Simplify the notation \( \left( {{\mathbb{L}}_{V\left( \alpha \right) },{d}_{\alpha }}\right) \) to \( \left( {{L}_{\alpha },{d}_{\alpha }}\right) \). Then \( {C}^{ * }\left( {{L}_{\alpha },{d}_{\alpha }}\right) \) is a Sullivan model for \( {X}_{\alpha } \). Now the Example of \( §{12}\left( \mathrm{c}\right) \) exhibits the fibre product \( \mathop{\prod }\limits_{{\alpha ⫫ k}}\left( {{C}^{ * }\left( {{L}_{\alpha },{d}_{\alpha }}\right) }\right) \) as a commutative model for \( X \). On the other hand the inclusions \( \left( {{L}_{\alpha },{d}_{\alpha }}\right) \rightarrow \mathop{\coprod }\limits_{\alpha }\left( {{L}_{\alpha },{d}_{\alpha }}\right) \) induce a dga morphism \[ {C}^{ * }\left( {\mathop{\coprod }\limits_{\alpha }\left( {{L}_{\alpha },{d}_{\alpha }}\right) }\right) \rightarrow \mathop{\prod }\limits_{\alpha }{}_{Ik}\left( {{C}^{ * }\left( {{L}_{\alpha },{d}_{\alpha }}\right) }\right) . \] We show that this is a quasi-isomorphism, thereby exhibiting \( \mathop{\coprod }\limits_{\alpha }\left( {{L}_{\alpha },{d}_{\alpha }}\right) \) as a Lie model for \( \mathop{\bigvee }\limits_{\alpha }{X}_{\alpha } \). To see that this is a quasi-isomorphism note that it is the dual of the horizontal arrow in  in which the slant arrows are the quasi-isomorphisms of Proposition 22.8. Finally, observe that the homology of a free product is the free product of the homologies, so that \[ {\pi }_{ * }\left( {\Omega \mathop{\bigvee }\limits_{\alpha }{X}_{\alpha }}\right) \otimes \mathbb{Q} = H\left( {\mathop{\coprod }\limits_{\alpha }{\mathbb{L}}_{V\left( \alpha \right) },{d}_{\alpha }}\right) = \mathop{\coprod }\limits_{\alpha }{\pi }_{ * }\left( {\Omega {X}_{\alpha }}\right) . \]
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Yes
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Example 3 The direct sum of Lie models is a Lie model of the product.
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Again let \( \left( {{L}_{\alpha },{d}_{\alpha }}\right) \) be Lie models for simply connected spaces \( {X}_{\alpha } \) such that \( X = \mathop{\prod }\limits_{\alpha }{X}_{\alpha } \) has rational homology of finite type. Then only finitely many \( {X}_{\alpha } \) will have rational homology degrees less than any given \( n \) and so we may suppose only finitely many \( {L}_{\alpha } \) have elements in degrees \( \leq n \) . It follows that\n\n\[ \n{C}^{ * }\left( {{\bigoplus }_{\alpha }\left( {{L}_{\alpha },{d}_{\alpha }}\right) }\right) = {\bigotimes }_{\alpha }{C}^{ * }\left( {{L}_{\alpha },{d}_{\alpha }}\right) ,\n\]\n\nwhich exhibits \( {\bigoplus }_{\alpha }\left( {{L}_{\alpha },{d}_{\alpha }}\right) \) as a Lie model for \( X \) (Example 2, \( §{12}\left( \mathrm{a}\right) \) ).
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Yes
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Example 4 A Lie model for \( {S}_{a}^{3} \vee {S}_{b}^{3}{ \cup }_{{\left\lbrack \alpha ,{\left\lbrack \alpha ,\beta \right\rbrack }_{W}\right\rbrack }_{W}}{D}^{8} \) .
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Let \( \alpha ,\beta \in {\pi }_{3}\left( {{S}_{a}^{3} \vee {S}_{a}^{3}}\right) \) be the elements represented by \( {S}_{a}^{3} \) and \( {S}_{b}^{3} \) respectively. Then (cf. Example 1) a Lie model for \( {S}_{a}^{3} \vee {S}_{b}^{3} \) is just \( \left( {\mathbb{L}\left( {v, w}\right) ,0}\right) \) with \( \deg v = \) \( \deg w = 2 \) and \( v, w \) corresponding to \( \alpha ,\beta \) . Moreover, the isomorphism \( {\pi }_{ * }\left( {{S}_{a}^{3} \vee }\right. \) \( \left. {S}_{b}^{3}\right) \otimes \mathbb{Q} \cong s\mathbb{L}\left( {v, w}\right) \) identifies \( {\left\lbrack \alpha ,{\left\lbrack \alpha ,\beta \right\rbrack }_{W}\right\rbrack }_{W} \) with \( s\left\lbrack {v,\left\lbrack {v, w}\right\rbrack }\right\rbrack \), as is shown in \( §{24}\left( \mathrm{\;b}\right) \) . Hence by Theorem 24.7,\n\n\[ \left( {\mathbb{L}\left( {v, w, u}\right) ,{du} = \left\lbrack {v\left\lbrack {v, w}\right\rbrack }\right\rbrack }\right) \text{is a Lie model for}{S}_{a}^{3} \vee {S}_{b}^{3}{ \cup }_{{\left\lbrack \alpha ,{\left\lbrack \alpha ,\beta \right\rbrack }_{W}\right\rbrack }_{W}}{D}^{8}\text{.} \]
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Yes
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Example 5 Lie models for \( \mathbb{C}{P}^{\infty } \) and \( \mathbb{C}{P}^{n} \) .
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The space \( \mathbb{C}{P}^{\infty } \) is a \( K\left( {\mathbb{Z},2}\right) \) and so its minimal Sullivan model is \( A = \left( {\Lambda \left( a\right) ,0}\right) \) with \( \deg a = 2 - \) cf. \( §{15}\left( \mathrm{\;b}\right) \), Example 2. Thus \( {\mathcal{L}}_{A} \) is a Lie model for \( \mathbb{C}{P}^{\infty } \) . This Lie model has the form \( \mathbb{L}\left( {{v}_{1},{v}_{2},{v}_{3},\ldots }\right) \) with \( {v}_{i} \) the desuspended dual of \( {a}^{i} \) . (Thus \( \deg {v}_{i} = {2i} - 1 \) ). In the coalgebra \( C \) dual to \( A \) let \( {c}_{i} \) be the element dual to \( {a}^{i} \) . Then \( \bar{\Delta }{c}_{k} = \mathop{\sum }\limits_{{i + j = k}}{c}_{i} \otimes {c}_{j} \) . Thus the formula in \( §{22}\left( \mathrm{e}\right) \) shows that the differential in \( {\mathcal{L}}_{A} \) is given by\n\n\[ d{v}_{k} = \frac{1}{2}\mathop{\sum }\limits_{{i + j = k}}\left\lbrack {{v}_{i},{v}_{j}}\right\rbrack \]\n\nSince \( H\left( {{\mathcal{L}}_{A}, d}\right) \cong {\pi }_{ * }\left( {\mathbb{C}{P}^{\infty }}\right) \otimes \mathbb{Q} \) it follows that\n\n\[ {H}_{n}\left( {{\mathcal{L}}_{A}, d}\right) = \left\{ \begin{array}{ll} \mathbb{Q} &, n = 2 \\ 0 & ,\text{ otherwise,} \end{array}\right. \]\n\n a fact that may not be immediately obvious from the formula for \( d \) .\n\nNotice that the construction of a CW-complex for this Lie model \( \left( {§{24}\left( \mathrm{e}\right) }\right) \) has one cell in each even degree \( {2n} \) and no cells of odd degree. The \( {2n} \) -skeleton has the rational homotopy type of \( \mathbb{C}{P}^{n} \), with Lie model \( \mathcal{L}\left( {{v}_{1},\ldots ,{v}_{n}}\right) \) and the same differential.
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Yes
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Example 6 A Lie model for \( \left( {\mathbb{C}{P}^{2} \vee {S}^{3}}\right) { \cup }_{{\left\lbrack \alpha ,\beta \right\rbrack }_{W}}{D}^{8} \) .
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As we saw in Example 5, a Lie model for \( \mathbb{C}{P}^{2} \) is just \( \left( {\mathbb{L}\left( {{v}_{1},{v}_{2}}\right), d}\right) \) with \( d{v}_{2} = \frac{1}{2}\left\lbrack {{v}_{1},{v}_{2}}\right\rbrack \) . Thus \( \left\lbrack {{v}_{1},{v}_{2}}\right\rbrack \) is a cycle and the homology class of \( s\left\lbrack {{v}_{1},{v}_{2}}\right\rbrack \) corresponds to a class \( \alpha \in {\pi }_{5}\left( {\mathbb{C}{P}^{2}}\right) \otimes \mathbb{Q} \) . Let \( \beta \in {\pi }_{3}\left( {S}^{3}\right) \) be the fundamental class. Then as in Example 4, Theorem 24.7 shows that\n\n\[ \left( {\mathbb{L}\left( {{v}_{1},{v}_{2}, w, u}\right) ;d{v}_{2} = \frac{1}{2}\left\lbrack {{v}_{1},{v}_{2}}\right\rbrack ,{dw} = 0,{du} = \left\lbrack {\left\lbrack {{v}_{1},{v}_{2}}\right\rbrack, w}\right\rbrack }\right) \]\n\nis a Lie model for \( \left( {\mathbb{C}{P}^{2} \vee {S}^{3}}\right) { \cup }_{{\left\lbrack \alpha ,\beta \right\rbrack }_{W}}{D}^{8} \) .
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Yes
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Proposition 24.8 When \( p \) is a Lie representative for a continuous map \( f \) : \( X \rightarrow Y \) between simply connected spaces, then \( \left( {I,{d}_{I}}\right) \) is a Lie model for the homotopy fibre of \( f \) .
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proof: By hypothesis there is a homotopy commutative diagram \n\n(24.9)\n\nNow let \( H \) be the homotopy from \( {m}_{X}{C}^{ * }\left( p\right) \) to \( {A}_{PL}\left( f\right) {m}_{Y} \) . Then in the commutative diagram \n\nwe may lift \( {m}_{X} \) through \( {id} \otimes {\varepsilon }_{0} \) because \( {C}^{ * }\left( p\right) \) is the inclusion of a relative Sullivan algebra (Proposition 14.4). This provides a homotopy \( {m}_{X} \sim {m}_{X}^{\prime } \) such that \( {m}_{X}^{\prime }{C}^{ * }\left( p\right) = {A}_{PL}\left( f\right) {m}_{Y} \) . In other words we may suppose that (24.9) commutes exactly.\n\nNow Proposition 15.5 identifies \( {C}^{ * }\left( {I,{d}_{I}}\right) \) as a Sullivan model for the homotopy fibre, \( F \), of \( f \), and hence by definition identifies \( \left( {I,{d}_{I}}\right) \) as a Lie model for \( F \) . \( ▱ \)
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Yes
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Proposition 24.8 When \( p \) is a Lie representative for a continuous map \( f \) : \( X \rightarrow Y \) between simply connected spaces, then \( \left( {I,{d}_{I}}\right) \) is a Lie model for the homotopy fibre of \( f \) .
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proof: By hypothesis there is a homotopy commutative diagram \n\n(24.9)\n\nNow let \( H \) be the homotopy from \( {m}_{X}{C}^{ * }\left( p\right) \) to \( {A}_{PL}\left( f\right) {m}_{Y} \) . Then in the commutative diagram \n\nwe may lift \( {m}_{X} \) through \( {id} \otimes {\varepsilon }_{0} \) because \( {C}^{ * }\left( p\right) \) is the inclusion of a relative Sullivan algebra (Proposition 14.4). This provides a homotopy \( {m}_{X} \sim {m}_{X}^{\prime } \) such that \( {m}_{X}^{\prime }{C}^{ * }\left( p\right) = {A}_{PL}\left( f\right) {m}_{Y} \) . In other words we may suppose that (24.9) commutes exactly.\n\nNow Proposition 15.5 identifies \( {C}^{ * }\left( {I,{d}_{I}}\right) \) as a Sullivan model for the homotopy fibre, \( F \), of \( f \), and hence by definition identifies \( \left( {I,{d}_{I}}\right) \) as a Lie model for \( F \) . \( ▱ \)
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Yes
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Lemma 26.2 \( \; \) Suppose \( \;\eta : \left( {B, d}\right) \overset{ \simeq }{ \rightarrow }\left( {A, d}\right) \; \) is a chain algebra quasi-isomorphism. Then\n\n(i) For any chain algebra morphism \( \varphi : \left( {{TV}, d}\right) \rightarrow \left( {A, d}\right) \) there is a morphism \( \psi : \left( {{TV}, d}\right) \rightarrow \left( {B, d}\right) \) such that \( \varphi \sim {\eta \psi } \) .\n\n(ii) If \( \psi ,{\psi }^{\prime } : \left( {{TV}, d}\right) \rightarrow \left( {B, d}\right) \) are morphisms satisfying \( {\eta \psi } \sim \eta {\psi }^{\prime } \) then \( \psi \sim \) \( {\psi }^{\prime } \) .
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proof: (i) We construct \( \psi \) and a homotopy \( \Phi : \varphi \sim {\eta \psi } \) by induction. Suppose they are defined in \( T\left( {V}_{ < n}\right) \) and choose a basis \( \left\{ {v}_{\alpha }\right\} \) for \( {V}_{n} \) . Then the cycles \( {\psi d}{v}_{\alpha } \) satisfy \( \eta \left( {{\psi d}{v}_{\alpha }}\right) = {\varphi d}{v}_{\alpha } - {d\Phi d}{v}_{\alpha } = d\left( {\varphi {v}_{\alpha } - {\Phi d}{v}_{\alpha }}\right) \) . Since \( \eta \) is a quasi-isomorphism there are elements \( {b}_{\alpha } \in B \) and \( {a}_{\alpha } \in A \) such that \( d{b}_{\alpha } = {\psi d}{v}_{\alpha } \) and \( \eta {b}_{\alpha } = \varphi {v}_{\alpha } - {\Phi d}{v}_{\alpha } + d{a}_{\alpha } \) . Extend \( \psi \) and \( \Phi \) to \( {V}_{n} \) by setting \( \psi {v}_{\alpha } = {b}_{\alpha } \) and \( \Phi {v}_{\alpha } = {a}_{\alpha } \) . Then \( \psi \) extends uniquely to an algebra morphism from \( T{V}_{ \leq n} \) and \( {\psi d} = {d\psi } \) because this relation holds in \( {V}_{ \leq n} \) . Similar \( \Phi \) extends uniquely to a \( \left( {\varphi ,{\eta \psi }}\right) \) -derivation from \( T{V}_{ \leq n} \) to \( B \) and \( \varphi - {\eta \psi } = {d\Phi } + {\Phi d} \) because this relation holds in \( {V}_{ \leq n} \) .\n\n(ii) By hypothesis there is an \( \left( {{\eta \psi },\eta {\psi }^{\prime }}\right) \) -derivation \( \Phi \) such that \( {\eta \psi } - \) \( \eta {\psi }^{\prime } = {d\Phi } + {\Phi d} \), and we seek to construct a \( \left( {\psi ,{\psi }^{\prime }}\right) \) -derivation \( \Psi \) such that \( \psi - {\psi }^{\prime } = \) \( {d\Psi } + {\Psi d} \) . Assume \( \Psi \) constructed in \( {V}_{ < n} \) so that \( {\eta \Psi } - \Phi = {d\Xi } - {\Xi d} \) for some\n\n\( \left( {{\eta \psi },\eta {\psi }^{\prime }}\right) \) -derivation \( \Xi \) . Again let \( \left\{ {v}_{\alpha }\right\} \) be a basis of \( {V}_{n} \) . Then \( \left( {\psi - {\psi }^{\prime }}\right) {v}_{\alpha } - {\Psi d}{v}_{\alpha } \) is a cycle \( {z}_{\alpha } \) in \( B \), and \( \eta {z}_{\alpha } = {d\Phi }{v}_{\alpha } - \left( {{\eta \Psi } - \Phi }\right) d{v}_{\alpha } = d\left( {\Phi {v}_{\alpha } - {\Xi d}{v}_{\alpha }}\right) \) . Choose \( {b}_{\alpha } \in B \) and \( {a}_{\alpha } \in A \) so that \( {z}_{\alpha } = d{b}_{\alpha } \) and \( \eta {b}_{\alpha } = \Phi {v}_{\alpha } - {\Xi d}{v}_{\alpha } + d{a}_{\alpha } \) . Then extend \( \Psi \) and \( \Xi \) by setting \( \Psi {v}_{\alpha } = {b}_{\alpha } \) and \( \Xi {v}_{\alpha } = {a}_{\alpha } \) .
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Yes
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Proposition 26.4 The maps \( {\mu }_{\text{alg }} \) and \( {\Delta }_{\text{alg }} \) make \( {C}_{ * }\left( G\right) \) into a differential graded Hopf algebra with identity and augmentation given by\n\n\[ \mathbf{k} = {C}_{ * }\left( {\{ c\} }\right) \rightarrow {C}_{ * }\left( G\right) \;\text{ and }\;{C}_{ * }\left( G\right) \rightarrow {C}_{ * }\left( {pt}\right) = \mathbf{k}. \]
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proof: This is a straightforward calculation using properties (4.4)-(4.9). In particular, the fact that \( {\Delta }_{\text{alg }} \) is a morphism of chain algebras follows from the compatibility (4.9) of \( {AW} \) and \( {EZ} \) .
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No
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Theorem 26.5 Let \( X \) be a simply connected topological space with rational homology of finite type. The choice of a free Lie model \( \left( {{\mathbb{L}}_{V}, d}\right) \) for \( X \) determines a natural homotopy class of chain algebra quasi-isomorphisms
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\[ \Theta : U\left( {{\mathbb{L}}_{V}, d}\right) \overset{ \simeq }{ \rightarrow }{C}_{ * }\left( {\Omega X}\right) \] such that \( \left( {\Theta \otimes \Theta }\right) {\Delta }_{U} \) and \( {\Delta }_{\mathrm{{alg}}} \circ \Theta \) are dga homotopic. Moreover \( H\left( \Theta \right) \) is the isomorphism (26.1).
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Yes
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Proposition 26.6 The map \( {\int }_{ * } : {C}_{ * }\langle \Gamma \rangle \rightarrow \left( {U{\mathbb{L}}_{V}, d}\right) \) is a chain algebra quasi-isomorphism.
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proof: Proposition 17.16 asserts that \( {\int }_{ * } \) is a quasi-isomorphism. Proposition 17.17 asserts that\n\n \( {C}_{ * }\langle \Gamma \otimes \Gamma \rangle \)\n\ncommutes. But the multiplication \( \left| {\Delta }_{\Gamma }\right| \) in \( \left| \Gamma \right| \) is the realization of \( {\Delta }_{\Gamma } \) which is the dual of the multiplication \( {\mu }_{U} \) in \( U{\mathbb{L}}_{V} \) . Thus for \( \sigma ,\tau \in \langle \Gamma \rangle \) ,\n\n\[ \n{\int }_{ * }\sigma \cdot \tau = {\int }_{ * }{C}_{ * }\left( {\Delta }_{\Gamma }\right) {EZ}\left( {\sigma \otimes \tau }\right) = {\mu }_{U}{\int }_{ * }{EZ}\left( {\sigma \otimes \tau }\right) = {\int }_{ * }\sigma \cdot {\int }_{ * }\tau .\n\]\n\nRecall now (§21(c)) that the universal enveloping algebra \( \left( {U{\mathbb{L}}_{V}, d}\right) \) is, as a chain algebra, the tensor algebra \( \left( {{TV}, d}\right) \) . Thus we may apply Lemma 26.2 to the chain algebra quasi-isomorphism \( {\int }_{ * } : {C}_{ * }\langle \Gamma \rangle \rightarrow \left( {{TV}, d}\right) \) to obtain a chain algebra quasi-isomorphism\n\n\[ \n\theta : \left( {U{\mathbb{L}}_{V}, d}\right) \overset{ \simeq }{ \rightarrow }{C}_{ * }\langle \Gamma \rangle \n\]\n\nuniquely determined up to dga homotopy by the requirement that \( {\int }_{ * } \circ \theta \sim {id} \) .
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Yes
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Lemma 26.7 The quasi-isomorphism \( \theta \) commutes with the comultiplications in \( U{\mathbb{L}}_{V} \) and in \( {C}_{ * }\langle \Gamma \rangle \) up to dga homotopy.
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proof: We have to show that\n\n\[ \left( {\theta \otimes \theta }\right) {\Delta }_{U} \sim {AW} \circ {C}_{ * }\left( {\Delta }_{\text{top }}\right) \circ \theta . \]\n\nNow here \( {\Delta }_{\text{top }} \) is just \( \left\langle {\mu }_{\Gamma }\right\rangle \), where \( {\mu }_{\Gamma } \) is the dual of \( {\Delta }_{U} \) . Consider the (noncommutative) diagram \n\nObserve first that\n\n\[ {C}_{ * }\langle \Gamma \rangle \otimes {C}_{ * }\langle \Gamma \rangle \overset{EZ}{ \rightarrow }{C}_{ * }\langle \Gamma \otimes \Gamma \rangle \overset{AW}{ \rightarrow }{C}_{ * }\langle \Gamma \rangle \otimes {C}_{ * }\langle \Gamma \rangle \]\nare dga morphisms, as follows easily from property (4.7) and (4.8) for EZ and from property (4.9) for \( {AW} \) . Moreover (Proposition 4.10) these are quasi-isomorphisms, and \( {AW} \circ {EZ} = {id} \) . In particular,\n\n\[ {AW} \circ {C}_{ * }\left\langle {\mu }_{\Gamma }\right\rangle \circ \theta = {\Delta }_{\text{alg }} \circ \theta = {AW} \circ {EZ} \circ {\Delta }_{\text{alg }} \circ \theta . \]\n\nSince \( {AW} \) is a chain algebra quasi-isomorphism we may apply Lemma 26.2(ii) to conclude that \( {C}_{ * }\left\langle {\mu }_{\Gamma }\right\rangle \circ \theta \sim {EZ} \circ {\Delta }_{\text{alg }} \circ \theta \) .\n\nFinally, note that the right hand square in the diagram above commutes by naturality. Thus\n\n\[ {\Delta }_{U} \sim {\Delta }_{U} \circ {\int }_{ * } \circ \;\theta \sim {\int }_{ * } \circ {C}_{ * }\langle {\mu }_{\Gamma }\rangle \circ \theta \sim {\int }_{ * } \circ {EZ} \circ {\Delta }_{\mathrm{{alg}}} \circ \theta \;. \]\n\nThe diagram in the proof of Proposition 26.6 allows us to replace \( {\int }_{ * } \circ {EZ} \) by \( {\int }_{ * } \otimes {\int }_{ * } \):\n\n\[ {\Delta }_{U} \sim {\int }_{ * } \otimes {\int }_{ * } \circ {\Delta }_{\text{alg }} \circ \theta \]\n\nThus, since \( {\int }_{ * }\theta \sim {id} \),\n\n\[ {\int }_{ * } \otimes {\int }_{ * } \circ \theta \otimes \theta \circ {\Delta }_{U} \sim {\Delta }_{U} \sim {\int }_{ * } \otimes {\int }_{ * } \circ {\Delta }_{\mathrm{{alg}}} \circ \theta . \]\n\nBut \( {\int }_{ * } \) is a quasi-isomorphism. Apply Lemma 26.2 to obtain\n\n\[ \theta \otimes \theta \circ {\Delta }_{U} \sim {\Delta }_{\text{alg }} \circ \theta . \]
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Yes
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Lemma 27.1 Suppose \( f : X \rightarrow Y \) is a continuous map.\n\n(i) If \( {f}^{\prime } \sim f \) then \( \operatorname{cat}{f}^{\prime } = \operatorname{cat}f \) .
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proof: (i) and (ii) are trivial consequences of the definitions. For (iii) suppose \( {g}^{\prime } \) is a homotopy inverse for \( g \) . Then \( {g}^{\prime }{gf} \sim f \) and so cat \( f = \operatorname{cat}{g}^{\prime }{gf} \leq \operatorname{cat}{gf} \leq \) cat \( f \) . Similarly if \( {f}^{\prime } \) is a homotopy inverse for \( f \) then \( \operatorname{cat}g = \operatorname{cat}{gf}{f}^{\prime } \leq \operatorname{cat}{gf} \leq \) cat \( g \) . (iv) follows from (iii) applied to \( i{d}_{Y} \circ f \) and \( f \circ i{d}_{X} \) and (v) is immediate from (i), (ii) and (iv).
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No
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Lemma 27.3 For any continuous map \( g : A \rightarrow X \) ,
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\[ \operatorname{cat}\left( {X{ \cup }_{g}{CA}}\right) \leq \operatorname{cat}X + 1. \] proof: Let \( \operatorname{cat}X = m \) and put \( \bar{a} = \left\lbrack {A\times \{ 0\} }\right\rbrack \in {CA} \) . Then \( X{ \cup }_{g}{CA} - \{ \bar{a}\} \) is an open subset of \( X{ \cup }_{g}{CA} \) containing \( X \) as a deformation retract. Hence \( \operatorname{cat}\left( {X{ \cup }_{g}{CA}-\{ \bar{a}\} }\right) = m \) and \( X{ \cup }_{g}{CA} - \{ \bar{a}\} = \mathop{\bigcup }\limits_{0}^{m}{U}_{i} \) with \( {U}_{i} \) open and contractible in \( X{ \cup }_{g}{CA} \) . Since \( {CA} - A \) is open and contractible in \( X{ \cup }_{g}{CA} \) it follows that \( \operatorname{cat}\left( {X{ \cup }_{g}{CA}}\right) \leq m + 1 \) .
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Yes
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Proposition 27.4 [158] Suppose \( \\left( {X,\\bar{x}}\\right) \) is path connected\n\n(i) If \( X \) is normal then Wh \( \\operatorname{cat}X \\leq \\operatorname{cat}X \) .\n\n(ii) If \( \\bar{x} \) is contained in a subspace \( U \) that is open and contractible in \( X \) then \( \\operatorname{cat}X \\leq \) Wh \( \\operatorname{cat}X \) .
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proof: (i) Let \( m = \\operatorname{cat}X \) so that \( X = \\mathop{\\bigcup }\\limits_{0}^{m}{U}_{i} \) with \( {U}_{i} \) open and contractible in \( X \) .\n\nBecause \( X \) is normal there are subspaces \( {A}_{i} \\subset {O}_{i} \\subset {B}_{i} \\subset {U}_{i} \) with \( {A}_{i},{B}_{i} \) closed and \( {O}_{i} \) open and \( X = \\mathop{\\bigcup }\\limits_{i}{A}_{i} \), and there are continuous functions \( {h}_{i} : X \\rightarrow I \) such that \( {\\left. {h}_{i}\\right| }_{{A}_{i}} \\equiv 1 \) and \( {\\left. {h}_{i}\\right| }_{X - {B}_{i}} \\equiv 0 \) .\n\nLet \( {H}_{i} : {U}_{i} \\times I \\rightarrow X \) be a homotopy from the inclusion of \( {U}_{i} \) to a constant map. Because \( X \) is path connected we may suppose \( {H}_{i}\\left( {-,1}\\right) : {U}_{i} \\rightarrow \\bar{x} \) . Define \( {K}_{i} : X \\times I \\rightarrow X \) by\n\n\[ \n{K}_{i}\\left( {x, t}\\right) = \\left\\{ \\begin{array}{lll} x & , & x \\in X - {B}_{i} \\\\ {H}_{i}\\left( {x,{h}_{i}\\left( x\\right) t}\\right) & , & x \\in {U}_{i} \\end{array}\\right. \n\]\n\nThen \( {K}_{i}\\left( {-,1}\\right) : {A}_{i} \\rightarrow \\bar{x} \) and so\n\n\[ \nK : X \\times I \\rightarrow {X}^{m + 1},\\;K\\left( {x, t}\\right) = \\left( {{K}_{0}\\left( {x, t}\\right) ,\\ldots ,{K}_{m}\\left( {x, t}\\right) }\\right) \n\]\n\nis a homotopy from \( \\Delta \) to a map \( X \\rightarrow {T}^{m + 1}\\left( X\\right) \) .\n\n(ii) Let \( m = \\operatorname{Wh}\\operatorname{cat}\\left( X\\right) \) and let \( K\\left( {x, t}\\right) = \\left( {{K}_{0}\\left( {x, t}\\right) ,\\ldots ,{K}_{m}\\left( {x, t}\\right) }\\right) \) be a homotopy from \( \\Delta \) to a map \( f = \\left( {{f}_{0},\\ldots ,{f}_{m}}\\right) : X \\rightarrow {T}^{m + 1}\\left( X\\right) \) . Set \( {U}_{i} = \) \( {f}_{i}^{-1}\\left( U\\right) \) . Then \( X = \\mathop{\\bigcup }\\limits_{0}^{m}{U}_{i} \) . Moreover \( {K}_{i} \) is a homotopy from the inclusion of \( {U}_{i} \) to the map \( {f}_{i} : {U}_{i} \\rightarrow U \) . Since \( U \) is contractible in \( X \) it follows that \( {U}_{i} \) is contractible in \( X \) too.
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Yes
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Proposition 27.5 Suppose \( X \) is an \( \left( {r - 1}\right) \) -connected CW complex of dimension \( d \) (some \( r \geq 1 \) ). Then\n\n\[ \operatorname{cat}X \leq d/r \]
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proof: It follows from the construction of Theorem 1.4 that there is a weak homotopy equivalence \( g : Y\overset{ \simeq }{ \rightarrow }X \) where \( Y \) is a CW complex whose \( \left( {r - 1}\right) \) - skeleton is a single 0-cell, \( {y}_{0} \) . By Corollary 1.7 this is a homotopy equivalence, and hence has a homotopy inverse \( f : X\overset{ \simeq }{ \rightarrow }Y \) .\n\nLet \( m \) be the integer part of \( d/r \), and recall from Example 3, \( §1 \) (a) that \( {Y}^{m + 1} \) is a CW complex whose \( k \) -cells are the products \( {D}_{{\alpha }_{0}}^{{k}_{0}} \times \cdots \times {D}_{{\alpha }_{m}}^{{k}_{m}} \) of cells in \( Y \) with \( \sum {k}_{i} = k \) . Since \( d < \left( {m + 1}\right) r \), the \( d \) -skeleton of \( {Y}^{m + 1} \) is contained in \( {T}^{m + 1}\left( Y\right) \) . In particular, a cellular approximation (Theorem 1.2) of the map \( {\Delta }_{f} = \left( {f,\ldots, f}\right) : X \rightarrow {Y}^{m + 1} \) is a map \( h : X \rightarrow {T}^{m + 1}\left( Y\right) \) such that \( {\Delta }_{f} \sim h \) .\n\nFinally, since \( {gf} \sim {id}, g \times \cdots \times g \circ {\Delta }_{f} \sim {\Delta }_{X},{\Delta }_{X} \) the diagonal of \( X \) . Thus \( {\Delta }_{X} \sim \left( {g \times \cdots \times g}\right) h : X \rightarrow {T}^{m + 1}\left( X\right) \) . By the Corollary to Proposition 27.4, \( \operatorname{cat}X \leq m \) .
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Yes
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Proposition 27.8 (Ganea [62]) The following conditions are equivalent on a continuous map \( f : Y \rightarrow X \) from a normal space \( Y \) :\n\n(i) \( f = {p}_{m}\sigma \) for some continuous \( \sigma : Y \rightarrow {P}_{m}X \) .\n\n(ii) \( f \sim {p}_{m}\sigma \) for some continuous \( \sigma : Y \rightarrow {P}_{m}X \) .\n\n(iii) \( \operatorname{cat}f \leq m \) .
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proof: \( \;\left( i\right) \Leftrightarrow \left( {ii}\right) \) : Suppose \( q : E \rightarrow X \) is any fibration and \( {h}_{E} : W \rightarrow E \) , \( {h}_{X} : W \rightarrow X \) are arbitrary continuous maps such that \( q{h}_{E} \sim {h}_{X} \) . Lift the homotopy starting at \( {h}_{E} \) to obtain \( {h}_{E} \sim h \) with \( {qh} = {h}_{X} \) . In particular (i) \( \Leftrightarrow \) (ii).\n\n(ii) \( \Rightarrow \) (iii): If (ii) holds apply Lemma 27.1 to conclude cat \( f = \) \( \operatorname{cat}{p}_{m}\sigma \leq \operatorname{cat}{p}_{m} \leq \operatorname{cat}{P}_{m}X \) . Since \( \operatorname{cat}{P}_{m}X \leq m \) by (27.7) it follows that \( \operatorname{cat}f \leq m \) .\n\n(iii) \( \Rightarrow \) (ii): Denote the constant map by \( {c}_{Y} : Y \rightarrow \left\{ {x}_{0}\right\} \) . If cat \( f = 0 \) then \( f \sim {c}_{Y} \), which certainly factors through \( {p}_{0} \) . Suppose cat \( f \leq m \), some \( m \geq 1 \) . Then \( Y = \mathop{\bigcup }\limits_{{i = 0}}^{m}{U}_{i} \), where the \( {U}_{i} \) are open and \( {\left. f\right| }_{{U}_{i}} \sim {\left. {c}_{Y}\right| }_{{U}_{i}} \) . Since \( Y \) is normal there are open subspaces \( {V}_{0},{V}_{1} \subset Y \) such that \( Y = {V}_{0} \cup {V}_{1} \) and such that the closures \( A \) and \( B \) of \( {V}_{0} \) and \( {V}_{1} \) satisfy \( A \subset \mathop{\bigcup }\limits_{{i = 0}}^{{m - 1}}{U}_{i} \) and \( B \subset {U}_{m} \) . Denote \( {\left. f\right| }_{A} \) and \( {\left. {c}_{Y}\right| }_{A} \) by \( {f}_{A} \) and \( {c}_{A} \) . Clearly \( \operatorname{cat}{f}_{A} \leq m - 1 \) . Since \( A \) is closed it is normal and so we may suppose by induction (because (i) \( \Leftrightarrow \) (ii)) that \( {f}_{A} = {p}_{m - 1}{\sigma }_{A} \) for some \( {\sigma }_{A} : A \rightarrow {P}_{m - 1}X \) . On the other hand, since \( B \subset {U}_{m} \) there is a homotopy \( {H}_{B} : B \times I \rightarrow {\left. X\text{ from }f\right| }_{B} \) to \( {\left. {c}_{Y}\right| }_{B}. \)
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Yes
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Theorem 27.11 (Cornea [40]) If \( X \) is a normal topological space then \( \operatorname{cat}X \leq m \Leftrightarrow X \vee {\sum Y} \) has the homotopy type of an \( m \) -cone for some \( m - 1 \) connected space \( Y \) .
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## proof:\n\nStep 1: Reduction to the case that \( B \) is an \( n \) -cone.\n\nLet \( g : Z \rightarrow B \) be a homotopy equivalence from an \( n \) -cone \( Z \) with constituent spaces \( {Y}_{k} \) . Then \( \left( {Z,{z}_{0}}\right) \) is well-based. Since \( B \) is path connected we may replace \( g \) with a homotopic based map; i.e. we may assume \( g : \left( {Z,{z}_{0}}\right) \rightarrow \left( {B,{b}_{0}}\right) \) .\n\nIn the pullback diagram\n\n\n\n\n\n\( {g}_{E} \) is a homotopy equivalence because \( g \) is (Proposition 2.3). By our second ’Basic Fact’ above, \( \left( {Z{ \times }_{B}E}\right) { \cup }_{j}{CF}\overset{ \simeq }{ \rightarrow }E{ \cup }_{j}{CF} \) ; i.e. we may assume \( B = Z \) is an \( n \) -cone with constituent spaces \( {Y}_{k} \) .\n\nWrite \( B = W{ \cup }_{f}\bar{C}A \) where \( \left( {W,{w}_{0}}\right) \) is an \( \left( {n - 1}\right) \) -cone with constituent spaces \( {Y}_{k}, k < n,\left( {A,{a}_{0}}\right) = \left( {{\sum }^{n}{Y}_{n},{y}_{n}}\right) \) and \( f : \left( {A,{a}_{0}}\right) \rightarrow \left( {W,{w}_{0}}\right) \) . Let \( q : {E}_{W} \rightarrow W \) be the restriction of the fibration \( q \) to \( W \) .\n\nStep 2: There is a homotopy equivalence of the form\n\n\[ {E}_{W}{ \cup }_{\theta }\left( {\bar{C}A \times F}\right) \overset{ \simeq }{ \rightarrow }E \]\n\nin which \( \theta : A \times F \rightarrow {E}_{W} \) restricts to \( j \) in \( \left\{ {a}_{0}\right\} \times F \) .\n\nLet \( \left( {\varphi, f}\right) : \left( {\bar{C}A, A}\right) \rightarrow \left( {B, W}\right) \) be the canonical map, and use it to pull the original fibration back to a pair of fibrations \n\nnoting that \( \left( {{\varphi }_{E},{f}_{E}}\right) \) is projection on \( \left( {E,{E}_{W}}\right) \) and \( \left( {p,{p}_{A}}\right)
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Yes
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Proposition 27.13 If \( B \) is normal then\n\n\[ \operatorname{cl}\left( {E{ \cup }_{j}{CF}}\right) \leq \operatorname{cl}B\;\text{ and }\;\operatorname{cat}\left( {E{ \cup }_{j}{CF}}\right) \leq \operatorname{cat}B. \]
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proof: The first assertion is immediate from Proposition 27.12. For the second, let \( \operatorname{cat}B = m \) . Then \( B \) is a homotopy retract of an \( m \) -cone \( P \) (Theorem 27.10). If \( B\overset{f}{ \rightarrow }P\overset{r}{ \rightarrow }B \) satisfy \( {rf} \sim i{d}_{B} \) then the homotopy lifts to a homotopy \( h \sim i{d}_{E} \) such that \( {qh} = {rf} \) . Thus we have maps of fibrations\n\n\n\nwhich exhibit \( E \cup {CF} \) as a homotopy retract of \( \left( {P{ \times }_{B}E}\right) \cup {CF} \) . Since \( P \) is an \( m \) -cone so is \( \left( {P{ \times }_{B}E}\right) \cup {CF} \), by Proposition 27.12, and so cat \( E \cup {CF} \leq m \) . \( ▱ \)
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Yes
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The LS category of the Ganea spaces \( {P}_{n}X \) .
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Suppose \( X \) is a normal topological space with \( \operatorname{cat}X = m, m < \infty \) . We show that\n\n\[ \operatorname{cat}{P}_{n}X = \left\{ \begin{array}{ll} n & \text{ if }n \leq m \\ m & \text{ if }n \geq m \end{array}\right.\]\n\nIn fact, since \( {P}_{n + 1}X \simeq {P}_{n}X \cup C{F}_{n} \) it follows that \( \operatorname{cat}{P}_{n + 1}X \leq \operatorname{cat}{P}_{n}X + \) 1 (Lemma 27.3). On the other hand, \( X \) is a retract of \( {P}_{m}X \) (Corollary to Proposition 27.8) and thus cat \( {P}_{m}X \geq \operatorname{cat}X = m \) . Since \( {P}_{0}X \) is contractible cat \( {P}_{0}X = 0 \) and now the first inequality implies cat \( {P}_{n}X = n, n \leq m \) .\n\nOn the other hand, Proposition 27.13 gives \( \operatorname{cat}{P}_{n}X \leq \operatorname{cat}X \), all \( n \) . Thus since \( X \) is a retract of \( {P}_{n}X, n \geq m \) we have \( \operatorname{cat}X \leq \operatorname{cat}{P}_{n}X \leq \operatorname{cat}X, n \geq m \) . \( ▱ \)
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Yes
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Proposition 27.14 If \( X \) is a path connected normal space then for any coefficient ring \( \mathbb{k} \) , \[ \mathrm{c}\left( {X;\mathbb{k}}\right) \leq \mathrm{e}\left( {X;\mathbb{k}}\right) \leq \operatorname{cat}X \leq \operatorname{cl}X. \]
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proof: To show \( \mathrm{c}\left( {X;\mathbb{k}}\right) \leq \mathrm{e}\left( {X;\mathbb{k}}\right) \) it is enough to show that \( \mathrm{c}\left( {Z;\mathbb{k}}\right) \leq n \) for \( n \) -cones \( Z \) . Write \( Z = Y{ \cup }_{h}\bar{C}A \) for some \( \left( {n - 1}\right) \) cone \( Y \), choose classes \( {\alpha }_{0},\ldots ,{\alpha }_{n} \in {H}^{ + }\left( {Z;\mathbb{R}}\right) \) and let \( \beta = {\alpha }_{0} \cup \cdots \cup {\alpha }_{n - 1} \) . Recall (§5) that the inclusion \( {C}^{ * }\left( {Z, Y;\mathbb{k}}\right) \rightarrow {C}^{ * }\left( {Z;\mathbb{k}}\right) \) induces a morphism \( \lambda : {H}^{ * }\left( {Z, Y;\mathbb{k}}\right) \rightarrow {H}^{ * }\left( {Z;\mathbb{k}}\right) \) of \( {H}^{ * }\left( {Z;\mathbb{R}}\right) \) -modules. Now by induction \( \beta \) restricts to zero in \( {H}^{ * }\left( {Y;\mathbb{R}}\right) \) and so \( \beta = \lambda {\beta }^{\prime } \), some \( {\beta }^{\prime } \in {H}^{ * }\left( {Z, Y;\mathbb{k}}\right) \) . Hence \( \beta \cup {\alpha }_{n} = \lambda \left( {{\beta }^{\prime } \cup {\alpha }_{n}}\right) \) . On the other hand, by excision the obvious map \( \left( {\varphi, h}\right) : \left( {{CA}, A}\right) \rightarrow \left( {Z, Y}\right) \) induces an isomorphism \( {H}^{ * }\left( {\varphi, h}\right) : {H}^{ * }\left( {Z, Y;\mathbb{R}}\right) \overset{ \cong }{ \rightarrow }{H}^{ * }\left( {{CA}, A;\mathbb{R}}\right) \) . Moreover \( {H}^{ * }\left( {\varphi, h}\right) \left( {{\beta }^{\prime } \cup {\alpha }_{n}}\right) = {H}^{ * }\left( {\varphi, h}\right) {\beta }^{\prime } \cup {H}^{ * }\left( \varphi \right) {\alpha }_{n} = 0 \), because \( \bar{C}A \) is contractible. Thus \( {\beta }^{\prime } \cup {\alpha }_{n} = 0 \) and hence so is \( \beta \cup {\alpha }_{n} = \lambda \left( {{\beta }^{\prime } \cup {\alpha }_{n}}\right) \) . This proves that \( \mathrm{c}\left( {Z;\mathbb{k}}\right) \leq n \) . The remaining inequalities are trivial consequences of Theorem 27.10 (where, indeed, the third inequality is stated).
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Yes
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Proposition 27.15 If \( \left( {X,{x}_{0}}\right) \) is well-based and normal then \( \mathrm{e}\left( {X;\mathbb{k}}\right) \) is the least integer \( m \) (or \( \infty \) ) such that \( {H}^{ * }\left( {{p}_{m};\mathbb{k}}\right) \) is injective.
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proof: Clearly \( \mathrm{e}\left( {X;\mathbb{R}}\right) \) is less than or equal to this least integer, because \( {P}_{n}X \) has the homotopy type of an \( n \) -cone (Proposition 27.9). On the other hand, if \( \mathrm{e}\left( {X;\mathbb{k}}\right) = m \) choose a map \( f : Z \rightarrow X \) from an \( m \) -cone such that \( {H}^{ * }\left( {f;\mathbb{k}}\right) \) is injective. We can suppose \( f \) is a based map because \( Z \) is well-based. (Replace \( f \) by a homotopic map if necessary.)\n\nRecall (§27(c)) that \( f \) induces continuous maps \( {P}_{m}f : {P}_{m}Z \rightarrow {P}_{m}X \) such that \( {p}_{m}^{X} \circ {P}_{m}f = f \circ {p}_{m}^{X} \) . Moreover, since \( Z \) is an \( m \) -cone cat \( Z \leq m \) (Theorem 27.10) and so there is a continuous map \( \sigma : Z \rightarrow {P}_{m}Z \) such that \( {p}_{m}^{Z} \circ \sigma = i{d}_{Z} \) . Then \( f = f \circ {p}_{m}^{Z} \circ \sigma = {p}_{m}^{X} \circ {P}_{m}f \circ \sigma \) . Since \( {H}^{ * }\left( {f;\mathbb{R}}\right) \) is injective so is \( {H}^{ * }\left( {{p}_{m}^{X};\mathbb{R}}\right) \) .
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Yes
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Proposition 27.16 Suppose \( X \) and \( Y \) are normal and simply connected, and let \( \mathbf{k} \subset \mathbb{Q} \) . If \( {H}^{ * }\left( {f;\mathbf{k}}\right) \) is an isomorphism then \( \mathrm{e}\left( {Y;\mathbf{k}}\right) = \mathrm{e}\left( {X;\mathbf{k}}\right) \) .
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proof of Proposition 27.16: If \( g : W \rightarrow Z \) is a continuous map and if \( {H}_{ * }\left( {g;\mathbb{R}}\right) \) is an isomorphism then \( {C}_{ * }\left( {g;\mathbb{R}}\right) \) is a chain equivalence (i.e., has a chain inverse) and so \( {C}^{ * }\left( {g;\mathbb{R}}\right) \) is a quasi-isomorphism too. In particular, \( {H}^{ * }\left( {f;\mathbb{R}}\right) \) and \( {H}^{ * }\left( {{P}_{m}f;\mathbb{k}}\right) \) are isomorphisms. Now use the construction \( X{ \cup }_{x}\left\lbrack {0,1}\right\rbrack \) to reduce to the case \( f \) is a based map between well-based spaces. Then apply Proposition 27.15.
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No
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Lemma 27.17 Suppose \( \mathbf{k} \subset \mathbb{Q} \) and \( {H}_{ * }\left( {f;\mathbf{k}}\right) \) is an isomorphism. Then \( {H}_{ * }\left( {{P}_{m}f;\mathbb{k}}\right) \) is an isomorphism for \( m \geq 1 \) . In particular if \( f \) is a weak homotopy equivalence so is each \( {P}_{m}f \) .
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proof: Suppose first that if \( {h}_{A} : {A}^{\prime } \rightarrow A \) and \( {h}_{B} : {B}^{\prime } \rightarrow B \) are continuous maps between path connected spaces, and that \( {H}_{ * }\left( {{h}_{A};\mathbb{k}}\right) \) and \( {H}_{ * }\left( {{h}_{B};\mathbb{k}}\right) \) are isomorphisms. Then the natural homeomorphism \( \left( {{CA} \times B}\right) { \times }_{A \times B}\left( {A \times {CB}}\right) \cong \) \( A * B \) of \( §1\left( \mathrm{f}\right) \) implies that \( {H}_{ * }\left( {{h}_{A} * {h}_{B};\mathbb{R}}\right) \) is an isomorphism.\n\nLet \( {F}_{m}f : {F}_{m}Y \rightarrow {F}_{m}X \) denote the restriction of \( {P}_{m}f \) to the fibres. Proposition 27.6 provides homotopy equivalences \( {F}_{m}\left( -\right) \simeq {F}_{m - 1}\left( -\right) * \Omega \left( -\right) \) which, with a little care, can be chosen to identify \( {F}_{m}f \) with \( {F}_{m - 1}f * {\Omega f} \) up to homotopy. The Whitehead-Serre theorem 8.6 asserts that \( {H}_{ * }\left( {{\Omega f};\mathbb{R}}\right) \) is an isomorphism. Hence so is each \( {H}_{ * }\left( {{F}_{m}f;\mathbb{k}}\right) \) .\n\nA simple van Kampen argument shows that the join of path connected spaces is simply connected. In particular, it follows from Theorem 8.6 that each \( {\pi }_{ * }\left( {{F}_{m}f}\right) \otimes \) \( \mathbb{k}, m \geq 1 \), is an isomorphism. Hence \( {P}_{m}X \) is simply connected, \( m \geq 1 \) and \( {\pi }_{ * }\left( {{P}_{m}f}\right) \otimes \mathbb{k} \) is an isomorphism (long exact homotopy sequence). A second application of Theorem 8.6 gives that \( {H}_{ * }\left( {{P}_{m}f;\mathbb{R}}\right) \) is an isomorphism. The final assertion of the lemma follows immediately from the same theorem.
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Yes
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Proposition 28.1 If \( X \) is a simply connected \( {CW} \) complex, then\n\n(i) \( {\operatorname{cat}}_{0}X = \operatorname{cat}{X}_{\mathbb{Q}} \) .\n\n(ii) \( {e}_{0}X = e\left( {X;\mathbb{Q}}\right) = e\left( {{X}_{\mathbb{Q}};\mathbb{Q}}\right) \) .
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proof of Proposition 28.1: Suppose \( X{ \simeq }_{\mathbb{Q}}Y \) . Choose a weak homotopy equivalence \( Z \rightarrow Y \) from a CW complex \( Z \) . Then \( {X}_{\mathbb{Q}} \) and \( {Z}_{\mathbb{Q}} \) have the same weak homotopy type. But these are CW complexes and so they have the same homotopy type. Hence \( \operatorname{cat}{X}_{\mathbb{Q}} = \operatorname{cat}{Z}_{\mathbb{Q}} \leq \operatorname{cat}Z \leq \operatorname{cat}Y \) .\n\nOn the other hand, Proposition 27.14 states that \( e\left( {Y;\mathbb{Q}}\right) = e\left( {Z;\mathbb{Q}}\right) \) and Proposition 27.15 identifies this as the least integer \( m \) such that \( {H}^{ * }\left( {{p}_{m}^{Z};\mathbb{Q}}\right) \) is injective. Rationalizing \( {P}_{m}Z \) and \( Z \), we can replace \( {p}_{m}^{Z} \) with \( {\left( {p}_{m}^{Z}\right) }_{\mathbb{Q}} \) . But it follows from the proof of Lemma 28.2(ii) that if \( {H}^{ * }\left( {{\left( {p}_{m}^{Z}\right) }_{\mathbb{Q}};\mathbb{Q}}\right) \) is injective so is \( {H}^{ * }\left( {{p}_{m}^{{Z}_{\mathbb{Q}}};\mathbb{Q}}\right) \) . Thus \( e\left( {{X}_{\mathbb{Q}};\mathbb{Q}}\right) = e\left( {{Z}_{\mathbb{Q}};\mathbb{Q}}\right) \leq e\left( {Z;\mathbb{Q}}\right) = e\left( {Y;\mathbb{Q}}\right) \) .
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Yes
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Lemma 28.2 Let \( X \) be a simply connected \( {CW} \) complex.\n\n(i) If \( f : X \rightarrow Y \) is a weak homotopy equivalence then \( \operatorname{cat}X \leq \operatorname{cat}Y \) .
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proof: (i) Put \( \operatorname{cat}Y = m \) . Then \( \operatorname{cat}f \leq m \) (Lemma 27.1). Hence \( f \) factors as \( {p}_{m}^{Y} \circ \sigma \) for some continuous \( \sigma : X \rightarrow {P}_{m}Y \) (Proposition 27.8). Since \( {P}_{m}f \) : \( {P}_{m}X \rightarrow {P}_{m}Y \) is also a weak homotopy equivalence (Lemma 27.17) we may lift \( \sigma \) through \( {P}_{m}f \) to construct a continuous \( \tau : X \rightarrow {P}_{m}X \) such that \( {P}_{m}f \circ \tau \sim \sigma \) . Then \( f \circ {p}_{m}^{X} \circ \tau \sim f \) and so \( {p}_{m}^{X} \circ \tau \sim i{d}_{X} \) (Lemma 1.4). It follows (Proposition 27.8) that \( \operatorname{cat}X \leq m \) .
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Yes
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Lemma 28.4 If \( \varrho : X \rightarrow Q \) is a rational homotopy equivalence to a simply connected \( r \) -cone, \( Q \), then there is a rational homotopy equivalence \( f : P \rightarrow X \) from a spherical \( r \) -cone, \( P \) .
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proof: Write \( Q = {Q}_{r} \supset \cdots \supset {Q}_{0} = \left\{ {q}_{0}\right\} \) with \( {Q}_{k + 1} = {Q}_{k}{ \cup }_{{g}_{k}}\bar{C}{\sum }^{k}{Y}_{k} \) as in \( §{27}\left( \mathrm{\;d}\right) \) .
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No
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Theorem 28.5 If \( X \) is a simply connected topological space then\n\n(i) \( {e}_{0}X \leq {\operatorname{cat}}_{0}X \leq {\operatorname{cl}}_{0}X \leq {\operatorname{cat}}_{0}X + 1 \) .
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proof: (i) Since the invariants are invariants of rational homotopy type we may suppose \( X \) is a CW complex, so that \( {e}_{0}X = e\left( {{X}_{\mathbb{Q}};\mathbb{Q}}\right) ,{\operatorname{cat}}_{0}X = \operatorname{cat}{X}_{\mathbb{Q}} \) and \( {\operatorname{cl}}_{0}X = \operatorname{cl}\left( {X}_{\mathbb{Q}}\right) \) . (Proposition 28.1 and 28.3). Now the first two inequalities of (i) follow from Proposition 27.14.\n\nFor the last inequality and the proof of (iii), let \( {\operatorname{cat}}_{0}X = m \) . We lose no generality in supposing \( X \) a rational CW complex. Then \( X \simeq {X}_{\mathbb{Q}} \) and cat \( X = m \) (Proposition 28.1). Thus the \( {m}^{\text{th }} \) Ganea fibration \( {p}_{m} : {P}_{m}X \rightarrow X \) admits a cross-section \( \sigma \) (Corollary to Proposition 27.8).\n\nConvert \( \sigma \) to a fibration \( E \rightarrow {P}_{m}X \) with fibre \( F \), and \( E \simeq X \) . In the proof of Theorem 27.11 we showed that\n\n\[ E \cup {CF} \simeq X \vee {\sum Y} \]\n\nwhere \( Y \) is a well-pointed space homotopy equivalent to \( F \) . Hence \( X \simeq (E \cup \) \( {CF}) \cup \overline{C}{\sum Y} \) .\n\nOn the other hand, we also showed in the proof of 27.11 that \( {\pi }_{ * }\left( F\right) \cong \) \( {\pi }_{* + 1}\left( {F}_{m}\right) ,{F}_{m} \) the fibre of \( {p}_{m} \) . But \( {F}_{m} \simeq {\sum }^{m}\left( {\Omega {X}^{\land m + 1}}\right) \) by Step 2 in the proof of Proposition 27.9. Since the smash of path connected well-based spaces is simply connected (trivial) it follows that \( F \) and \( Y \) are \( m \) -connected. Finally, Proposition 27.9 identifies \( {P}_{m}X \) as homotopy equivalent to an \( m \) -cone and so \( E \cup {CF} \) also has this property (Proposition 27.13). Since \( X \simeq \left( {E \cup {CF}}\right) \cup \bar{C}{\sum Y} \) , it follows that \( {\operatorname{cl}}_{0}X \leq \operatorname{cl}X \leq m + 1 \) .\n\nNote that we have identified \( X \) as a retract of \( {P}_{m}X \) which is homotopy equivalent to an \( m \) -cone.
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Yes
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Theorem 28.6 (Mapping theorem) Let \( f : X \rightarrow Y \) be a continuous map between simply connected topological spaces. If \( {\pi }_{ * }\left( f\right) \otimes \mathbb{Q} \) is injective then\n\n\[{\operatorname{cat}}_{0}X \leq {\operatorname{cat}}_{0}Y\]
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proof: We lose no generality in assuming \( X \) and \( Y \) are rational CW complexes. Let \( {\operatorname{cat}}_{o}Y = m \) and convert \( f \) to the fibration \( g : E = X{ \times }_{Y}{MY} \rightarrow Y \) as described in \( §2\left( \mathrm{c}\right) \) . Then inclusion of the fibre \( X{ \times }_{Y}{PY} \) induces zero in homotopy, because \( {\pi }_{ * }\left( g\right) \) is injective. The homotopy equivalence \( E \rightarrow X \) converts this inclusion to a fibration \( p : X{ \times }_{Y}{PY} \rightarrow X \) and so \( {\pi }_{ * }\left( p\right) = 0 \) and the fibre inclusion \( j : {\Omega Y} \rightarrow X{ \times }_{Y}{PY} \) satisfies: \( {\pi }_{ * }\left( j\right) \) is surjective.\n\nDecompose the rational vector spaces \( {\pi }_{n}\left( {\Omega Y}\right) \) as \( {\pi }_{n}^{\prime } \oplus {\pi }_{n}^{\prime \prime } \), with \( {\pi }_{ * }\left( j\right) : {\pi }_{n}^{\prime }\overset{ \cong }{ \rightarrow } \) \( {\pi }_{n}\left( {X{ \times }_{Y}{PY}}\right) \) . Let \( {K}_{n}^{\prime } \) and \( {K}_{n}^{\prime \prime } \) be cellular Eilenberg-MacLane spaces of types \( \left( {{\pi }_{n}^{\prime }, n}\right) \) and \( \left( {{\pi }_{n}^{\prime \prime }, n}\right) \), respectively. Apply the Corollary to Proposition 16.7 to obtain a weak homotopy equivalence\n\n\[ \varphi : \mathop{\prod }\limits_{n}^{ \sim }{K}_{n}^{\prime } \times \mathop{\prod }\limits_{n}^{ \sim }{K}_{n}^{\prime \prime } \rightarrow {\Omega Y} \]\n\nThen setting \( {K}^{\prime } = \mathop{\prod }\limits_{n}^{ \sim }{K}_{n}^{\prime } \) we see that \( {j\varphi } : {K}^{\prime } \rightarrow X{ \times }_{Y}{PY} \) is a weak homotopy equivalence.\n\nNow let \( \lambda : X{ \times }_{Y}{PY} \rightarrow E \) be the inclusion and set\n\n\[ \psi = \left( {{id},{Cj\varphi }}\right) : E{ \cup }_{\lambda j\varphi }C{K}^{\prime } \rightarrow E{ \cup }_{\lambda }C\left( {X{ \times }_{Y}{PY}}\right) . \]\n\nBy Proposition 27.13, \( \operatorname{cat}\left( {E \cup C\left( {X{ \times }_{Y}{PY}}\right) }\right) \leq m \) . Moreover, since \( {\lambda j} \) is the constant map, \( E{ \cup }_{\lambda j\varphi }C{K}^{\prime } = E \vee \left( {C{K}^{\prime }/{K}^{\prime }}\right) \), and this space is simply connected. Thus \( \psi \) is a weak homotopy equivalence, by an obvious homology calculation.\n\nSince \( E \vee \left( {C{K}^{\prime }/{K}^{\prime }}\right) \) has the homotopy type of a CW complex we conclude from Lemma 28.2 that\n\n\[ {\operatorname{cat}}_{0}X \leq \operatorname{cat}X \leq \operatorname{cat}\left( {E \vee \left( {C{K}^{\prime }/{K}^{\prime }}\right) }\right) \]\n\n\[ \leq \operatorname{cat}\left( {E{ \cup }_{\lambda }C\left( {X{ \times }_{Y}{PY}}\right) }\right) \leq m. \]
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Yes
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Let \( X \) be a simply connected CW complex and suppose \( {\pi }_{i}\left( X\right) = 0, i < r \) . From Proposition 4.20 we obtain a fibration \( X\overset{p}{ \rightarrow }K\left( {{\pi }_{r}\left( X\right), r}\right) \) such that \( {\pi }_{r}\left( p\right) \) is the identity. If \( Z \) is a CW complex mapping by a weak homotopy equivalence to the fibre of \( p \) then the composite \( Z\overset{f}{ \rightarrow }X \) satisfies: \( {\pi }_{i}\left( f\right) \) is an isomorphism for \( i \geq r + 1 \) . In this way we obtain a sequence of maps
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\[ \rightarrow {X}^{n + 1}\overset{{g}_{n}}{ \rightarrow }\cdots \rightarrow {X}^{3}\overset{{g}_{2}}{ \rightarrow }{X}^{2} = X \] such that \( {X}^{n + 1} \) is an \( n \) -connected CW complex and \( {\pi }_{i}\left( {g}_{n}\right) \) is an isomorphism for \( i \geq n + 1 \) . The \( {X}^{n} \) are called Postnikov fibres of \( X \) . Notice that we can apply the Mapping theorem to this sequence to obtain \[ \cdots \leq {\operatorname{cat}}_{0}{X}^{n} \leq \cdots \leq {\operatorname{cat}}_{0}{X}^{2} \leq {\operatorname{cat}}_{0}X. \] In particular, if \( X \) has finite rational LS category, so do all its Postnikov fibres.
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Yes
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Free loop spaces have infinite rational category.
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Let \( X \) be a topological space. The free loop space of \( X \) is the space \( {X}^{{S}^{1}} \) of all continuous maps \( {S}^{1} \rightarrow X\left( {§0}\right) \) . We show that\n\n- If \( X \) is two-connected and if \( {H}_{ + }\left( {X;\mathbb{Q}}\right) \neq 0 \) then\n\n\[{\operatorname{cat}}_{0}\left( {X}^{{S}^{1}}\right) = \infty\]\n\n(Better results can be proved with a little more work.)\n\nIndeed, let \( e \in {S}^{1} \) be a basepoint. Evaluation at \( e \) defines a fibration \( p \) : \( {X}^{{S}^{1}} \rightarrow X \) whose fibre at a basepoint \( {x}_{0} \in X \) is the loop space \( {\Omega X} \) . The map \( s : X \rightarrow {X}^{{S}^{1}} \) which associates to \( x \) the constant loop \( {S}^{1} \rightarrow x \) is a cross-section for \( p \), and it follows that the inclusions \( j : {\Omega X} \rightarrow {X}^{{S}^{1}} \) and \( s : X \rightarrow {X}^{{S}^{1}} \) are injective in homotopy.\n\nSuppose \( {\operatorname{cat}}_{0}\left( {X}^{{S}^{1}}\right) = m < \infty \) . Since \( X \) is 2-connected, \( {\Omega X} \) and \( {X}^{{S}^{1}} \) are simply connected. Thus Theorem 28.6 asserts that if \( {\operatorname{cat}}_{0}\left( {X}^{{S}^{1}}\right) = m < \infty \) , \( {\operatorname{cat}}_{0}X \leq m \) and \( {\operatorname{cat}}_{0}{\Omega X} \leq m \) . Suppose \( {f}_{i} : {S}^{2{n}_{i} + 1} \rightarrow X \) and \( {g}_{j} : {S}^{2{m}_{j}} \rightarrow X \) represent linearly independent elements in \( {\pi }_{ * }\left( X\right) \otimes \mathbb{Q} \) . Then, as in Proposition 16.7, \( \mathop{\prod }\limits_{i}\Omega {f}_{i} \times \mathop{\prod }\limits_{j}{\widehat{g}}_{j} : \mathop{\prod }\limits_{i}\Omega {S}^{2{n}_{i} + 1} \times \mathop{\prod }\limits_{j}{S}^{2{m}_{j}} \rightarrow {\Omega X} \) is injective in rational homotopy, and so the category of this product is bounded by \( m \) . Since \( {H}^{ * }\left( {\Omega {S}^{2{n}_{i} + 1};\mathbb{C}}\right) \) is a polynomial algebra (Example 1, \( §{15}\left( \mathrm{\;b}\right) \) ) it follows that \( {\pi }_{\text{odd }}\left( X\right) \otimes \mathbb{Q} = 0 \) and \( \dim {\pi }_{\text{even }}\left( X\right) \otimes \mathbb{Q} \) is finite. Now \( X \) has a Sullivan model of the form \( \Lambda {V}^{\text{even }} \) and so its cohomology is a polynomial algebra, contradicting \( {\operatorname{cat}}_{0}X < \infty \) .
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Yes
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Proposition 28.7 If \( \alpha \in {\left( {L}_{X}\right) }_{n} \) corresponds to a Gottlieb element in \( {\pi }_{n + 1}\left( X\right) \otimes \mathbb{Q} \) then \[ \left\lbrack {\alpha ,\beta }\right\rbrack = 0,\;\beta \in {L}_{X}. \]
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proof: Let \( \widehat{\alpha } \in {\pi }_{n + 1}\left( X\right) \otimes \mathbb{Q} \) be the Gottlieb element corresponding to \( \alpha \) and let \( \widehat{\alpha } \) be represented by a map \( f : \left( {{S}^{n + 1}, * }\right) \rightarrow \left( {X, * }\right) \) . Without loss of generality assume \( X \) is a rational CW complex. Then for any \( g : \left( {{S}^{k}, * }\right) \rightarrow \left( {X, * }\right) \) the map \( \left( {f, g}\right) : {S}^{n + 1} \vee {S}^{k} \rightarrow X \) extends to a map \( {S}^{n + 1} \times {S}^{k} \rightarrow X \), which shows that the Whitehead product of \( \widehat{\alpha } = \left\lbrack f\right\rbrack \) and \( \left\lbrack g\right\rbrack \) is zero (§13(d)).
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Yes
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Proposition 28.8 Suppose \( X \) is a simply connected topological space of finite rational LS category. Then\n\n(i) \( {G}_{ * }^{\mathbb{Q}}\left( X\right) \) is concentrated in odd degrees, and\n\n(ii) \( \dim {G}_{ * }^{\mathbb{Q}}\left( X\right) \leq {\operatorname{cat}}_{0}X \) .
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proof: (i) We may assume \( X \) itself is a simply connected rational CW complex, and hence that \( {G}_{ * }\left( X\right) = {G}_{ * }^{\mathbb{Q}}\left( X\right) \) . Suppose first that \( f : {S}^{2k} \rightarrow X \) represents a non-zero Gottlieb element, and that \( X \) is \( \left( {{2k} - 1}\right) \) -connected. By Theorem 4.19, \( {H}_{2k}\left( {X;\mathbb{Q}}\right) = {\pi }_{2k}\left( X\right) \otimes \mathbb{Q} = {\pi }_{2k}\left( X\right) \) . Thus there is a cohomology class \( w \in \) \( {H}^{2k}\left( {X;\mathbb{Q}}\right) \) such that \( {H}^{ * }\left( f\right) w \neq 0 \) . Extend \( \left( {f,\ldots, f, i{d}_{X}}\right) \) to a map \( \varphi : {S}^{2k} \times \) \( \cdots \times {S}^{2k} \times X \rightarrow X \) and observe that \( {H}^{ * }\left( \varphi \right) w = {H}^{ * }\left( f\right) w \otimes 1 \otimes \cdots \otimes 1 + \) \( \cdots + 1 \otimes \cdots \otimes {H}^{ * }\left( f\right) w \otimes 1 + 1 \otimes \cdots \otimes 1 \otimes w. \) If we have used \( n \) -factors \( {S}^{2k} \) then \( {H}^{ * }\left( \varphi \right) {w}^{n} = {\left( {H}^{ * }\left( \varphi \right) w\right) }^{n} \neq 0 \) . It follows that \( {w}^{n} \neq 0 \) for all \( n \) and hence \( {\operatorname{cat}}_{0}X = \infty \) .\n\nNow suppose \( X \) is only simply connected, and let \( {X}^{2k} \rightarrow X \) be its \( 2{k}^{\text{th }} \) Postnikov fibre. If \( f : {S}^{2k} \rightarrow X \) represents a non-zero element of \( {G}_{2k}\left( X\right) \) then we may assume \( f : {S}^{2k} \rightarrow {X}^{2k} \) . Apply the argument of Lemma 1.5 to the diagram\n\n\n\nto fill in the dotted arrow making the upper triangle commute. This identifies \( f \) as representing a non-zero element of \( {G}_{2k}\left( {X}^{2k}\right) \) .\n\nBut now the Mapping theorem 28.6 gives \( {\operatorname{cat}}_{0}X \geq {\operatorname{cat}}_{0}{X}^{2k} = \infty \) .\n\n(ii) Again we may suppose \( X \) is a simply connected rational CW complex. Thus \( {G}_{ * }^{\mathbb{Q}}\left( X\right) = {G}_{ * }\left( X\right) \) . If \( {\alpha }_{1},\ldots ,{\alpha }_{r} \) are linearly independent elements of \( {G}_{ * }\left( X\right) \) then, as at the start of this topic we can extend representatives \( {f}_{i} : {S}^{{n}_{i}} \rightarrow X \) to
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Yes
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Proposition 28.8 Suppose \( X \) is a simply connected topological space of finite rational LS category. Then\n\n(i) \( {G}_{ * }^{\mathbb{Q}}\left( X\right) \) is concentrated in odd degrees, and\n\n(ii) \( \dim {G}_{ * }^{\mathbb{Q}}\left( X\right) \leq {\operatorname{cat}}_{0}X \) .
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proof: (i) We may assume \( X \) itself is a simply connected rational CW complex, and hence that \( {G}_{ * }\left( X\right) = {G}_{ * }^{\mathbb{Q}}\left( X\right) \) . Suppose first that \( f : {S}^{2k} \rightarrow X \) represents a non-zero Gottlieb element, and that \( X \) is \( \left( {{2k} - 1}\right) \) -connected. By Theorem 4.19, \( {H}_{2k}\left( {X;\mathbb{Q}}\right) = {\pi }_{2k}\left( X\right) \otimes \mathbb{Q} = {\pi }_{2k}\left( X\right) \) . Thus there is a cohomology class \( w \in \) \( {H}^{2k}\left( {X;\mathbb{Q}}\right) \) such that \( {H}^{ * }\left( f\right) w \neq 0 \) . Extend \( \left( {f,\ldots, f, i{d}_{X}}\right) \) to a map \( \varphi : {S}^{2k} \times \) \( \cdots \times {S}^{2k} \times X \rightarrow X \) and observe that \( {H}^{ * }\left( \varphi \right) w = {H}^{ * }\left( f\right) w \otimes 1 \otimes \cdots \otimes 1 + \) \( \cdots + 1 \otimes \cdots \otimes {H}^{ * }\left( f\right) w \otimes 1 + 1 \otimes \cdots \otimes 1 \otimes w. \) If we have used \( n \) -factors \( {S}^{2k} \) then \( {H}^{ * }\left( \varphi \right) {w}^{n} = {\left( {H}^{ * }\left( \varphi \right) w\right) }^{n} \neq 0 \) . It follows that \( {w}^{n} \neq 0 \) for all \( n \) and hence \( {\operatorname{cat}}_{0}X = \infty \) .\n\nNow suppose \( X \) is only simply connected, and let \( {X}^{2k} \rightarrow X \) be its \( 2{k}^{\text{th }} \) Postnikov fibre. If \( f : {S}^{2k} \rightarrow X \) represents a non-zero element of \( {G}_{2k}\left( X\right) \) then we may assume \( f : {S}^{2k} \rightarrow {X}^{2k} \) . Apply the argument of Lemma 1.5 to the diagram\n\n\n\nto fill in the dotted arrow making the upper triangle commute. This identifies \( f \) as representing a non-zero element of \( {G}_{2k}\left( {X}^{2k}\right) \) .\n\nBut now the Mapping theorem 28.6 gives \( {\operatorname{cat}}_{0}X \geq {\operatorname{cat}}_{0}{X}^{2k} = \infty \) .\n\n(ii) Again we may suppose \( X \) is a simply connected rational CW complex. Thus \( {G}_{ * }^{\mathbb{Q}}\left( X\right) = {G}_{ * }\left( X\right) \) . If \( {\alpha }_{1},\ldots ,{\alpha }_{r} \) are linearly independent elements of \( {G}_{ * }\left( X\right) \) then, as at the start of this topic we can extend representatives \( {f}_{i} : {S}^{{n}_{i}} \rightarrow X \) to a map\n\n\[ \varphi : {S}^{{n}_{1}} \times \cdots \times {S}^{{n}_{r}} \times X \rightarrow X. \]\n\nLet \( g \) be the restriction of \( \varphi \) to \( {S}^{{n}_{1}} \times \cdots \times {S}^{{n}_{r}} \) . Then by (i) each \( {n}_{i} \) is odd and so \( {\pi }_{ * }\left( {S}^{{n}_{i}}\right) \otimes \mathbb{Q} = {\pi }_{{n}_{i}}\left( {S}^{{n}_{i}}\right) \otimes \mathbb{Q} = \mathbb{Q} \) (Example 1, \( §{15}\left( \mathrm{\;d} \)
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Yes
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Lemma 29.2 Suppose \( \left( {A, d}\right) \) is a commutative cochain algebra (over any \( \mathbf{k} \) of characteristic zero) such that \( {A}^{0} = \mathbb{k},{H}^{1}\left( A\right) = 0 \) and each \( {H}^{i}\left( A\right) \) has finite dimension. Then there is a subcochain algebra \( \left( {B, d}\right) \) such that: \( {B}^{1} = 0 \), each \( {B}^{i} \) is finite dimensional and \( d : {B}^{ + } \rightarrow {B}^{ + } \cdot {B}^{ + } \) .
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proof: Choose \( \widehat{A} \subset A \) so that \( {\widehat{A}}^{1} = 0,{\widehat{A}}^{2} \oplus d\left( {A}^{1}\right) = {A}^{2} \) and \( {\widehat{A}}^{i} = {A}^{i}, i \geq 3 \) . Then \( \left( {\widehat{A}, d}\right) \overset{ \simeq }{ \rightarrow }\left( {A, d}\right) \) and so we may suppose \( {A}^{1} = 0 \) .\n\nSuppose next that \( {A}^{i} \) is finite dimensional for \( i < n \), and let \( \bar{A} \) be the subalgebra generated by \( {A}^{ < n} \) and \( {\left( \operatorname{Im}d\right) }^{n} \) . Then \( \bar{A} \) is a sub cochain algebra and each \( {\bar{A}}^{i} \) is finite dimensional. Now write \( {A}^{n} = {d}^{-1}\left( {\bar{A}}^{n + 1}\right) \oplus {U}^{n} \) and choose \( C \subset A \) so that (i) \( {C}^{i} = {A}^{i}, i < n \) ,(ii) \( {C}^{n} = {d}^{-1}\left( {\bar{A}}^{n + 1}\right) \) ,(iii) \( {C}^{n + 1} \supset {\bar{A}}^{n + 1} \) and \( {C}^{n + 1} \oplus d{U}^{n} = {A}^{n + 1} \) and (iv) \( {C}^{i} = {A}^{i}, i \geq n + 2 \) . Then \( \left( {C, d}\right) \) is a sub cochain algebra including quasi-isomorphically in \( \left( {A, d}\right) \) and \( {C}^{i} \) is finite dimensional, \( i \leq n \) .\n\nWe now construct a decreasing sequence of quasi-isomorphic inclusions of sub cochain algebras \( A \supset A\left( 2\right) \supset A\left( 3\right) \supset \cdots \) as follows: if \( A\left( k\right) \) is constructed so that \( {A}^{i}\left( k\right) \) is finite dimensional, \( i \leq k \), then \( A\left( {k + 1}\right) \) is a subcochain algebra including quasi-isomorphically in \( A\left( k\right) \) such that \( {A}^{i}\left( {k + 1}\right) = {A}^{i}\left( k\right), i \leq k \) and such that among all these \( {A}^{k + 1}\left( {k + 1}\right) \) has minimal dimension (necessarily finite by the argument above). Set \( B = \mathop{\bigcap }\limits_{k}A\left( k\right) \) and note that \( {B}^{i} = {A}^{i}\left( k\right), i \leq k \), \n\nso that \( \left( {B, d}\right) \overset{ \simeq }{ \rightarrow }\left( {A, d}\right) \) . Note as well that for any proper sub cochain algebra \( \left( {\widehat{B}, d}\right) \subset \left( {B, d}\right) \) the inclusion is not a quasi-isomorphism.\n\nLet \( \zeta : B \rightarrow \mathbb{R} \oplus {B}^{ + }/{B}^{ + } \cdot {B}^{ + } \) be the projection and let \( Q\left( d\right) \) be the differential in \( {B}^{ + }/{B}^{ + } \cdot {B}^{ + } \) . If \( \ker Q\left( d\right) = H \oplus \operatorname{Im}Q\left( d\right) \) then \( {\zeta }^{-1}\left( {\mathbb{R} \oplus H}\right) \) is a sub cochain algebra including quasi-isomorphically in \( B \) . Thus \( {\zeta }^{-1}\left( {\mathbb{R} \oplus H}\right) = B \) and \( H = \) \( {B}^{ + }/{B}^{ + } \cdot {B}^{ + } \) ; i.e., \( Q\left( d\right) = 0 \) .
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Yes
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Corollary 1 Suppose for some \( n \geq r \geq 1 \) that the non-zero elements of \( {H}^{ + }\left( {{\Lambda V}, d}\right) \) are concentrated in degrees \( i \) with \( r \leq i \leq n \) . Then\n\n\[ \operatorname{cat}\left( {{\Lambda V}, d}\right) \leq n/r \]
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proof: It is easy to construct a quasi-isomorphism \( \left( {{\Lambda V}, d}\right) \overset{ \simeq }{ \rightarrow }\left( {B, d}\right) \) in which \( {B}^{ + } \) is concentrated in degrees \( i, r \leq i \leq n \) . Then nil \( B \leq n/r \) . Apply the Proposition.
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Yes
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Proposition 29.4 Suppose \( \left( {{\Lambda V}, d}\right) \) is a rational Sullivan model for a simply connected space \( X \) with rational homology of finite type. Then\n\n\[ \operatorname{cat}\left( {{\Lambda V}, d}\right) = {\operatorname{cat}}_{0}X\;\text{ and }\;e\left( {{\Lambda V}, d}\right) = {e}_{0}X. \]
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proof: We may suppose \( X \) is a rational CW complex, so that \( {\operatorname{cat}}_{0}X = \operatorname{cat}X \) (Proposition 28.1). Recall the quasi-isomorphism \( {\zeta }_{m} : \left( {{\Lambda V} \otimes {\Lambda Z}\left( m\right), d}\right) \overset{ \simeq }{ \rightarrow } \) \( \left( {{\Lambda V}/{\Lambda }^{ > m}V, d}\right) \) defined at the start of this section. The Sullivan algebra \( ({\Lambda V} \otimes \) \( {\Lambda Z}\left( m\right), d) \) is a Sullivan model for the realization \( Y = \left| {{\Lambda V} \otimes {\Lambda Z}\left( m\right), d}\right| \), as we showed in Theorem 17.10. Thus \( \left( {{\Lambda V}/{\Lambda }^{ > m}V, d}\right) \) is a commutative model for \( Y \), which implies that \( {\operatorname{cl}}_{0}\left( Y\right) \leq m \) (Theorem 29.1). Since \( Y \) is a rational CW complex, \( Y \) has the homotopy type of an \( m \) -cone (Proposition 28.3).\n\nSuppose \( \operatorname{cat}\left( {{\Lambda V}, d}\right) = m \) . Then there is a morphism \( {\pi }_{m} : \left( {{\Lambda V} \otimes {\Lambda Z}\left( m\right), d}\right) \rightarrow \) \( \left( {{\Lambda V}, d}\right) \) such that \( {\pi }_{m}{\lambda }_{m} = {id} \) . Thus \( \left| {\lambda }_{m}\right| \left| {\pi }_{m}\right| = i{d}_{\left| \Lambda V, d\right| } \) and \( \left| {{\Lambda V}, d}\right| \) is a retract of \( Y \) . Since \( \left| {{\Lambda V}, d}\right| { \simeq }_{\mathbb{Q}}X \) (Theorem 17.12) it follows that \( {\operatorname{cat}}_{0}X \leq \operatorname{cat}\left| {{\Lambda V}, d}\right| \leq m \).\n\nConversely, if \( {\operatorname{cat}}_{0}X = m \) then \( \operatorname{cat}X = m \) and \( X \) is a homotopy retract of an \( m \) -cone \( P \) (Theorem 27.10). Let \( \left( {{\Lambda V}, d}\right) \overset{\varphi }{ \rightarrow }\left( {\Lambda {V}_{P}, d}\right) \overset{\psi }{ \rightarrow }\left( {{\Lambda V}, d}\right) \) be Sullivan representatives respectively for the retraction and the inclusion. Then \( {\psi \varphi } \) is a quasi-isomorphism and hence an isomorphism (Theorem 14.11). Moreover, Theorem 29.1 provides a quasi-isomorphism \( \zeta : \left( {\Lambda {V}_{P}, d}\right) \overset{ \simeq }{ \rightarrow }\left( {A, d}\right) \) with nil \( A \leq \) \( m \) . Thus \( \operatorname{cat}\left( {{\Lambda V}, d}\right) \leq m \) (Proposition 29.3).\n\nFinally, since \( {e}_{0}X = e\left( {X;\mathbb{Q}}\right) \) this is the least integer, \( r \), such that there is a continuous map \( f \) of \( X \) into an \( r \) -cone such that \( {H}^{r}\left( {f;\mathbb{Q}}\right) \) is injective (Proposition 28.1). Now a simplified version of the argument above shows that \( e\left( {{\Lambda V}, d}\right) = {e}_{0}X \) .
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Yes
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Example 1 A space \( X \) satisfying \( {c}_{0}X < {e}_{0}X \) .
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Let \( X \) be a simply connected space with Sullivan model \( \left( {\Lambda \left( {x, y, z}\right), d}\right) \) where \( \deg x = 3 = \deg y,{dx} = {dy} = 0 \) and \( {dz} = {xy} \) . Then the cohomology algebra \( {H}^{ * }\left( {X;\mathbb{Q}}\right) \) has \( 1,\left\lbrack x\right\rbrack ,\left\lbrack y\right\rbrack ,\left\lbrack {xz}\right\rbrack ,\left\lbrack {yz}\right\rbrack \) and \( \left\lbrack x\right\rbrack \left\lbrack {yz}\right\rbrack \) as basis, and so\n\n\[ \n{c}_{0}X = 2\text{.}\n\]\n\nOn the other hand, \( \operatorname{nil}\left( {\Lambda \left( {x, y, z}\right) }\right) = 3 \) and so \( 3 \geq {\operatorname{cl}}_{0}X \geq {\operatorname{cat}}_{0}X \geq {e}_{0}X \) . Finally, \( {xyz} \in {\Lambda }^{3}\left( {x, y, z}\right) \) represents a non-trivial cohomology class, so \( {e}_{0}X \geq 3 \) . Thus\n\n\[ \n{e}_{0}X = {\operatorname{cat}}_{0}X = {\operatorname{cl}}_{0}X = 3.\n\]
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Yes
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A space \( X \) satisfying \( {e}_{0}X < {\operatorname{cat}}_{0}X \) .
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Consider the commutative cochain algebra \( \left( {A, d}\right) \) given by\n\n\[ A = \Lambda \left( {x, y, t}\right) /\left( {{x}^{4},{xy},{xt}}\right) ,\;{dx} = {dy} = 0,{dt} = {x}^{3}, \]\n\nwith \( \deg x = 2,\deg y = 3 \) and \( \deg t = 5 \) . Evidently\n\n\[ \operatorname{nil}\left( {A, d}\right) = 3. \]\n\nThus if \( X \) is a simply connected space with \( \left( {A, d}\right) \) as commutative model then\n\n\[ {\operatorname{cat}}_{0}X \leq {\operatorname{cl}}_{0}X = 3. \]\n\nA vector space basis for \( {A}^{ + } \) is given by \( x,{x}^{2},{x}^{3}, y, t,{yt} \) and so a vector space basis for \( {H}^{ + }\left( A\right) \) is given by \( \left\lbrack x\right\rbrack ,{\left\lbrack x\right\rbrack }^{2},\left\lbrack y\right\rbrack ,\left\lbrack {yt}\right\rbrack \) . In particular, \( H\left( A\right) \) is concentrated in degrees \( \leq 7 \) . A minimal Sullivan model \( \left( {{\Lambda V}, d}\right) \) for \( \left( {A, d}\right) \) has the form \( m \) : \( \Lambda \left( {x, y, z, t,\ldots }\right) \rightarrow A \) with \( {mx} = x,{my} = y,{mz} = 0,{mt} = t \) and \( {dz} = {xy} \) ; the remaining generators all have degree at least 7 . In particular \( {\Lambda }^{ \geq 3}V = \mathbb{k}z{x}^{2} \oplus \) \( {\left( {\Lambda }^{ \geq 3}V\right) }^{ > 7} \) . Since \( z{x}^{2} \) is not a cocycle no cohomology class can be represented by a cocycle in \( {\Lambda }^{ \geq 3}V \) :\n\n\[ {e}_{0}\left( X\right) = e\left( {{\Lambda V}, d}\right) = 2. \]\n\nFinally, we show that \( \operatorname{cat}\left( {{\Lambda V}, d}\right) > 2 \), thereby establishing\n\n\[ {\operatorname{cl}}_{0}\left( X\right) = {\operatorname{cat}}_{0}\left( X\right) = \operatorname{cat}\left( {{\Lambda V}, d}\right) = 3. \]\n\nThus in particular, \( {e}_{0}\left( X\right) < {\operatorname{cat}}_{0}\left( X\right) \) . In fact, let \( \varrho : \left( {{\Lambda V}, d}\right) \rightarrow \left( {{\Lambda V}/{\Lambda }^{ > 2}V, d}\right) \) be the projection. If \( \operatorname{cat}\left( {{\Lambda V}, d}\right) \leq 2 \) then there is a morphism of graded algebras \( \varphi : H\left( {{\Lambda V}/{\Lambda }^{ > 2}V}\right) \rightarrow H\left( {\Lambda V}\right) \) such that \( {\varphi H}\left( \varrho \right) = {id} \) . Now \( t \) becomes a cocycle in \( {\Lambda V}/{\Lambda }^{ > 2}V \) and so we would have \( \left\lbrack {{yt} + {x}^{2}z}\right\rbrack \overset{H\left( \varrho \right) }{ \mapsto }\left\lbrack y\right\rbrack \left\lbrack t\right\rbrack \overset{\varphi }{ \mapsto }0 \) . Since \( \left\lbrack {{yt} + {x}^{2}z}\right\rbrack \mapsto \) \( \left\lbrack {yt}\right\rbrack \) in \( H\left( A\right) ,\left\lbrack {{yt} + {x}^{2}z}\right\rbrack \neq 0 \) and this contradiction proves \( \operatorname{cat}\left( {{\Lambda V}, d}\right) > 2 \) .
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Yes
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Let \( X \) be a simply connected topological space with rational homology of finite type. If \( X \) is formal then
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\[ {c}_{0}\left( X\right) = {e}_{0}\left( X\right) = {\operatorname{cat}}_{0}X = {\operatorname{cl}}_{0}X. \] In fact, since \( X \) is formal \( {H}^{ * }\left( X\right) \) is a commutative model for \( X \) . Thus \( {c}_{0}\left( X\right) = \) nil \( {H}^{ * }\left( X\right) \geq {\operatorname{cl}}_{0}X \), by Theorem 29.1. The reverse inequalities are established in Theorem 28.5.
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Yes
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We show for coformal spaces that\n\n\[ \n{e}_{0}\left( X\right) = {\operatorname{cat}}_{0}\left( X\right) = {\operatorname{cl}}_{0}X \n\]
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In fact, suppose \( {e}_{0}\left( X\right) = r \) . Choose a vector space complement \( S \) for \( \ker {d}_{1} \) in \( {\Lambda }^{r}V \) . Then \( I = S \oplus {\Lambda }^{ > r}V \) is an acyclic ideal and \( \left( {{\Lambda V}/I, d}\right) \) is a commutative model for \( X \) . Since \( \operatorname{nil}\left( {{\Lambda V}/I, d}\right) = r \) we have (Theorem 29.1) that \( r \geq {\operatorname{cl}}_{0}X \) . The reverse inequalities are always true, as in Example 1.
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No
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Example 6 Minimal Sullivan algebras \( \left( {{\Lambda V}, d}\right) \) with \( V = {V}^{\text{odd }} \) and \( \dim V < \infty \) .
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If \( \left( {{\Lambda V}, d}\right) \) is as in the title of the example, let \( r = \dim V \) . Then \( {\Lambda V} = {\bigoplus }_{i = 0}^{r}{\Lambda }^{i}V \) , \( \dim {\Lambda }^{r}V = 1 \) and the elements in \( {\Lambda }^{r}V \) are cocycles and not coboundaries. Thus \( e\left( {{\Lambda V}, d}\right) = r = \operatorname{nil}\left( {{\Lambda V}, d}\right) \) and hence \( e\left( {{\Lambda V}, d}\right) = \operatorname{cat}\left( {{\Lambda V}, d}\right) = r \) .
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Yes
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Example 7 \( \left( {{\Lambda V}, d}\right) = \Lambda \left( {a, b, x, y, z}\right) \) with \( {dx} = {a}^{2},{dy} = {b}^{2} \) and \( {dz} = {ab} \) .
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In this example we take \( \deg a = 2,\deg b = 2 \), but any even degrees would do. Hence \( \left( {{\Lambda V}, d}\right) \overset{ \simeq }{ \rightarrow }{\Lambda a}/{a}^{2} \otimes {\Lambda b}/{b}^{2} \otimes {\Lambda z} \), and this commutative model \( \left( {A, d}\right) \) has nil \( A = 3 \) . A non trivial cohomology class is represented by \( {abz} - x{b}^{2} \) and so \( e\left( {{\Lambda V}, d}\right) \geq 3 \) . It follows that \( e\left( {{\Lambda V}, d}\right) = \operatorname{cat}\left( {{\Lambda V}, d}\right) = 3 \) .
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Yes
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Example 1 \( \;\mathrm{e}\left( {\left( {{\Lambda V}, d}\right) \otimes \left( {{\Lambda W}, d}\right) }\right) = \mathrm{e}\left( {{\Lambda V}, d}\right) + \mathrm{e}\left( {{\Lambda W}, d}\right) \) .
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Let \( \left( {{\Lambda V}, d}\right) \) and \( \left( {{\Lambda W}, d}\right) \) be any minimal Sullivan algebras. Then \( \mathrm{e}\left( {{\Lambda V}, d}\right) \) is the least integer \( r \) such that \( \left( {{\Lambda V}, d}\right) \rightarrow \left( {{\Lambda V}/{\Lambda }^{ > r}V, d}\right) \) is injective in cohomology. Thus it is the greatest integer \( r \) such that \( \Lambda {V}^{ \geq r} \) contains a cocycle \( v \) representing a non-zero class in \( H\left( {\Lambda V}\right) \) . Let \( s = \mathrm{e}\left( {{\Lambda W}, d}\right) \) and \( w \in {\Lambda }^{ \geq s}W \) be a cocycle representing a non-zero class in \( H\left( {\Lambda W}\right) \) . Then \( v \otimes w \in {\Lambda }^{ \geq r + s}\left( {V \otimes W}\right) \) and \( \left\lbrack {v \otimes w}\right\rbrack \neq 0 \), so \( \mathrm{e}\left( {\left( {{\Lambda V}, d}\right) \otimes \left( {{\Lambda W}, d}\right) }\right) \geq \) \( r + s \) . Conversely the surjection \( {\Lambda V} \otimes {\Lambda W} \rightarrow {\Lambda V}/{\Lambda }^{ > r}V \otimes {\Lambda W}/{\Lambda }^{ > s}W \) is injective in cohomology, and its kernel contains \( {\Lambda }^{ > r + s}\left( {V \oplus W}\right) \) . Thus \( \mathrm{e}\left( {\left( {{\Lambda V}, d}\right) \otimes \left( {{\Lambda W}, d}\right) }\right) \leq \) \( r + s \) ; i.e. \( \mathrm{e}\left( {\left( {{\Lambda V}, d}\right) \otimes \left( {{\Lambda W}, d}\right) }\right) = \mathrm{e}\left( {{\Lambda V}, d}\right) + \mathrm{e}\left( {{\Lambda W}, d}\right) \) .
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Yes
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Example 2 \( {\operatorname{cat}}_{0}X - {\mathrm{e}}_{0}X \) can be arbitrarily large.
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Let \( X \) be the space of Example 2, \( §{29} \) (b). Its minimal Sullivan model \( \left( {{\Lambda V}, d}\right) \) satisfies \( \operatorname{cat}\left( {{\Lambda V}, d}\right) = 3 \) and \( \mathrm{e}\left( {{\Lambda V}, d}\right) = 2 \), and has the form \( \Lambda \left( {x, y, z, t, u, v,\ldots }\right) \) with \( {dx} = {dy} = 0,{dz} = {xy},{dt} = {x}^{3},{du} = {zy},{dv} = {z}^{2} - {2ux} \) and with \( \deg x = 2 \) , \( \deg y = 3,\deg z = 4,\deg t = 5,\deg u = 6 \) and \( \deg v = 7 \) . Set \( u \) and \( y \) to zero to define a surjection\n\n\[ \left( {\Lambda \left( {x, y, z, t, u, v}\right), d}\right) \rightarrow \left( {\Lambda \left( {x, z, t, v}\right) ,\bar{d}}\right) \]\n\nwith \( \bar{d}x = \bar{d}z = 0,\bar{d}t = {x}^{3} \) and \( \bar{d}v = {z}^{2} \) ; denote this quotient Sullivan algebra by \( \left( {{\Lambda W},\bar{d}}\right) \) . Then \( H\left( {{\Lambda W},\bar{d}}\right) \) is concentrated in degrees \( \leq 8 \) while the remaining elements of \( V \) have degrees at least 8 . It follows that the surjection above extends to a surjective morphism \( \varphi : \left( {{\Lambda V}, d}\right) \rightarrow \left( {{\Lambda W},\bar{d}}\right) \) . Taking tensor products we obtain surjective morphisms\n\n\[ { \otimes }^{n}\varphi : {\left( \Lambda V, d\right) }^{\otimes n} \rightarrow {\left( \Lambda W,\bar{d}}\right) }^{\otimes n}. \]\n\nThe cocycle \( z{x}^{2} \) in \( {\Lambda }^{3}W \) represents a non trivial cohomology class. Thus \( \mathrm{e}\left( {{\Lambda W}, d}\right) \geq 3 \) and \( \mathrm{e}\left( {\left( \Lambda W, d\right) }^{\otimes n}\right) \geq {3n} \) (Example 1). Now apply the Mapping theorem 29.5 to obtain\n\n\[ \operatorname{cat}{\left( \Lambda V, d\right) }^{\otimes n} \geq \operatorname{cat}{\left( \Lambda W,\bar{d}}\right) }^{\otimes n} \geq \mathrm{e}\left( {{\Lambda W},\bar{d}}\right) \geq {3n}. \]\n\nOn the other hand, again by Example 1,\n\n\[ \mathrm{e}{\left( \Lambda V, d\right) }^{\otimes n} = n\mathrm{e}\left( {{\Lambda V}, d}\right) = {2n}. \]\n\nThus if \( {X}^{n} = X \times \cdots \times X \) ( \( n \) factors),\n\n\[ {\operatorname{cat}}_{0}\left( {X}^{n}\right) \geq {3n}\;\text{ and }\;{\mathrm{e}}_{0}\left( {X}^{n}\right) = {2n}. \]\n\n(In fact, it is not too hard to show that \( {\operatorname{cat}}_{0}\left( {X}^{n}\right) = {3n} \) .)
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Yes
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Example 1 A non-trivial Gottlieb element.\n\nLet \( \\left( {A, d}\\right) \) be the commutative cochain algebra defined by \( A = \\Lambda \\left( {a, b, x, y}\\right) /{abxy} \) with \( \\deg a = \\deg b = \\deg x = 3 \) and \( {dy} = {abx} \) . The Sullivan model for \( \\left( {A, d}\\right) \) has the form \( \\left( {{\\Lambda V}, d}\\right) = \\Lambda \\left( {a, b, x, y, z,\\ldots }\\right) \) with \( {dz} = {abxy} \) . We show that the map \( f : y \\mapsto 1 \) is a Gottlieb element for \( \\left( {{\\Lambda V}, d}\\right) \) .
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In fact extend \( f \) to a derivation \( \\theta \) of \( \\left( {{\\Lambda V}, d}\\right) \) by first setting \( {\\theta z} = {yx} \) . Then note that elements of higher degree in \( V \) have degree at least 15, while \( H\\left( {{\\Lambda V}, d}\\right) \) is concentrated in degrees \( \\leq {11} \) . Thus \( \\theta \) extends automatically to the rest of \( V \) so as to satisfy \( {\\theta d} + {d\\theta } = 0 \) .
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Yes
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Theorem 29.9 (Hess - [90]) Assume there is a morphism\n\n\\[ \n\\eta : \\left( {{\\Lambda V} \\otimes {\\Lambda Z}, d}\\right) \\rightarrow \\left( {{\\Lambda V}, d}\\right)\n\\]\n\nof \\( \\left( {{\\Lambda V}, d}\\right) \\) -modules such that \\( {\\eta \\lambda } = {id} \\) . Then there is a morphism \\( {\\eta }^{\\prime } : ({\\Lambda V} \\otimes \\) \\( {\\Lambda Z}, d) \\rightarrow \\left( {{\\Lambda V}, d}\\right) \\) of cochain algebras such that \\( {\\eta }^{\\prime }\\lambda = {id} \\) ; i.e.,\n\n\\[ \n\\operatorname{cat}\\left( {{\\Lambda V}, d}\\right) \\leq m.\n\\]
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proof of Theorem 29.9: Regard \\( \\left( {{\\Lambda V} \\otimes \\left( {\\mathbb{k} \\oplus {Z}_{0}}\\right), d}\\right) \\) as a \\( \\left( {{\\Lambda V}, d}\\right) \\) -bimodule by writing\n\n\\[ \n{\\Omega \\Phi } = {\\left( -1\\right) }^{\\deg \\Phi \\deg \\Omega }{\\Phi \\Omega } = {\\left( -1\\right) }^{\\deg \\Phi \\deg \\Omega }\\Phi \\land \\Omega ,\\;\\begin{array}{l} \\Phi \\in {\\Lambda V} \\\\ \\Omega \\in {\\Lambda V} \\otimes \\left( {\\mathbb{k} \\oplus {Z}_{0}}\\right) . \\end{array}\n\n\\]\n\nThen define a product in \\( {\\Lambda V} \\otimes \\left( {\\mathbb{R} \\oplus {Z}_{0}}\\right) \\) by setting\n\n\\[ \n{\\Omega }_{1} \\circ {\\Omega }_{2} = \\left( {\\eta {\\Omega }_{1}}\\right) {\\Omega }_{2} + {\\Omega }_{1}\\left( {\\eta {\\Omega }_{2}}\\right) - \\left( {\\eta {\\Omega }_{1}}\\right) \\left( {\\eta {\\Omega }_{2}}\\right) .\n\\]\n\nA simple calculation shows that this product is associative and makes \\( \\left( {{\\Lambda V} \\otimes \\left( {\\mathbb{k} \\oplus {Z}_{0}}\\right), d}\\right) \\) into a commutative cochain algebra. We denote this cochain algebra by \\( \\left( {A, d}\\right) \\) . It is immediate that \\( \\lambda : \\left( {{\\Lambda V}, d}\\right) \\rightarrow \\left( {A, d}\\right) \\) is a morphism of cochain algebras as is the restriction \\( {\\eta }_{A} : \\left( {A, d}\\right) \\rightarrow \\left( {{\\Lambda V}, d}\\right) \\) of \\( \\eta \\) .\n\nWe prove the theorem by constructing a morphism \\( \\varphi : \\left( {{\\Lambda V} \\otimes {\\Lambda Z}, d}\\right) \\rightarrow \\left( {A, d}\\right) \\) of cochain algebras such that \\( {\\varphi \\lambda } = \\lambda \\) . Then \\( {\\eta }^{\\prime } = {\\eta }_{A}\\varphi : \\left( {{\\Lambda V} \\otimes {\\Lambda Z}, d}\\right) \\rightarrow \\left( {{\\Lambda V}, d}\\right) \\) is the desired retraction.\n\nFirst observe (Proposition 29.10 (iii)) that \\( \\zeta \\) restricts to a surjective quasi-isomorphism \\( \\left( {{\\Lambda V} \\otimes \\left( {\\mathbb{R} \\oplus {Z}_{0}}\\right), d}\\right) \\rightarrow \\left( {{\\Lambda V}/{\\Lambda }^{ > m}V, d}\\right) \\) . Thus its kernel, \\( I \\), satisfies \\( H\\left( {I, d}\\right) = 0
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Yes
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Field extension preserves category.
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Let \( \left( {{\Lambda V}, d}\right) \) be a minimal Sullivan algebra with \( V = {\left\{ {V}^{i}\right\} }_{i > 2} \), and let \( \mathbb{K} \) be a field extension of \( \mathbb{R} \) . Then \( \left( {{\Lambda V}, d}\right) \otimes \mathbb{K} \) is a minimal Sullivan algebra over \( \mathbb{K} \) . We show now that\n\n\[ \operatorname{cat}\left( {{\Lambda V}, d}\right) = \operatorname{cat}\left( {\left( {{\Lambda V}, d}\right) \otimes \mathbb{K}}\right) . \]\n\nSuppose \( \operatorname{cat}\left( {\left( {{\Lambda V}, d}\right) \otimes \mathbb{K}}\right) = m \) and let\n\n\n\nbe morphisms of \( \left( {{\Lambda V}, d}\right) \otimes \mathbb{K} \) modules as in the Corollary to Theorem 29.9. In particular, \( {\Lambda }^{ > m}V \cdot M = 0 \) and \( {\beta \alpha } = {id} \) . Let \( \lambda : \mathbb{R} \rightarrow \mathbb{K} \) the inclusion and let \( \pi : \mathbb{K} \rightarrow \mathbb{k} \) be a \( \mathbb{k} \) -linear map such that \( {\pi \lambda } = {id} \) . Then the identity of \( \left( {{\Lambda V}, d}\right) \) factors as\n\n\[ \left( {{\Lambda V}, d}\right) \overset{\alpha \lambda }{ \rightarrow }\left( {P, d}\right) \overset{\pi \beta }{ \rightarrow }\left( {{\Lambda V}, d}\right) \]\n\nand \( \gamma \) is also a quasi-isomorphism of \( \left( {{\Lambda V}, d}\right) \) -modules. Thus the Corollary to Theorem 29.9 asserts that \( \operatorname{cat}\left( {{\Lambda V}, d}\right) \leq m \) .\n\nConversely, if \( \operatorname{cat}\left( {{\Lambda V}, d}\right) = m \) we can apply \( - \otimes \mathbb{K} \) to the diagram of the Corollary to conclude \( \operatorname{cat}\left( {\left( {{\Lambda V}, d}\right) \otimes \mathbb{K}}\right) \leq m \) .
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Yes
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Let \( M \) be a simply connected smooth manifold with rational cohomology of finite type, and let \( \left( {{\Lambda W}, d}\right) \rightarrow {A}_{DR}\left( M\right) \) be a minimal Sullivan model for the cochain algebra of \( {C}^{\infty } \) differential forms on \( M \) . We observe that \[ {\operatorname{cat}}_{0}M = \operatorname{cat}\left( {{\Lambda W}, d}\right) . \]
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In fact, as we saw in \( §{11},{A}_{DR}\left( M\right) \) is connected by quasi-isomorphisms of commutative cochain algebras to \( {A}_{PL}\left( {M;\mathbb{Q}}\right) \otimes \mathbb{R} \) . Thus if \( \left( {{\Lambda V}, d}\right) \) is a rational minimal Sullivan model for \( M \) then \( \left( {{\Lambda W}, d}\right) \cong \left( {{\Lambda V}, d}\right) \otimes \mathbb{R} \) . It follows now from Example 1 (and Proposition 29.4) that \( {\operatorname{cat}}_{0}M = {\operatorname{cat}}_{0}\left( {{\Lambda V}, d}\right) = \operatorname{cat}\left( {{\Lambda W}, d}\right) \) .
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Yes
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Proposition 29.15 For a minimal Sullivan algebra \( \left( {{\Lambda V}, d}\right) \) and any \( \left( {{\Lambda V}, d}\right) - \) module, \( \left( {M, d}\right) \) :\n\n(i) \( \mathrm{e}\left( {M, d}\right) \leq \operatorname{mcat}\left( {M, d}\right) \) .\n\n(ii) \( \operatorname{mcat}\left( {M, d}\right) \leq \operatorname{mcat}\left( {{\Lambda V}, d}\right) \) .\n\n(iii) If \( V = {\left\{ {V}^{i}\right\} }_{i \geq 2} \) then \( \operatorname{mcat}\left( {{\Lambda V}, d}\right) = \operatorname{cat}\left( {{\Lambda V}, d}\right) \) .
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proof: (i) This is immediate from the definitions.\n\n(ii) Suppose \( \operatorname{mcat}\left( {{\Lambda V}, d}\right) = k \) and construct a homotopy commutative diagram\n\n\n\nand a morphism \( \eta : \left( {N, d}\right) \rightarrow \left( {{\Lambda V}, d}\right) \), such that \( \zeta \) is a semifree resolution and \( {\eta \lambda } \sim {id} \) . Let \( \left( {P, d}\right) \) be a semifree resolution for \( \left( {M, d}\right) \), and apply \( - { \otimes }_{\Lambda V}P \) to this diagram. This yields\n\n\n\nwith \( {\eta }^{\prime }{\lambda }^{\prime } \sim {id} \) . It follows that \( \operatorname{mcat}\left( {M, d}\right) \leq k \) .\n\n(iii) Let \( \zeta : \left( {{\Lambda V} \otimes {\Lambda Z}, d}\right) \rightarrow \left( {{\Lambda V}/{\Lambda }^{ > m}V, d}\right) \) be a minimal Sullivan model for the surjection \( \varrho \) . It is in particular a \( \left( {{\Lambda V}, d}\right) \) -semifree resolution. If \( \operatorname{cat}\left( {{\Lambda V}, d}\right) = m \) there is a morphism \( \eta : \left( {{\Lambda V} \otimes {\Lambda Z}, d}\right) \rightarrow \left( {{\Lambda V}, d}\right) \) of Sullivan algebras restricting to the identity in \( \left( {{\Lambda V}, d}\right) \) . In particular, \( \eta \) is \( {\Lambda V} \) -linear and so \( \operatorname{mcat}\left( {{\Lambda V}, d}\right) \leq m \) .\n\nConversely, if \( \operatorname{mcat}\left( {{\Lambda V}, d}\right) = m \) then there is a morphism \( \eta : \left( {{\Lambda V} \otimes {\Lambda Z}, d}\right) \rightarrow \) \( \left( {{\Lambda V}, d}\right) \) of \( \left( {{\Lambda V}, d}\right) \) -modules such that \( {\eta \lambda } \sim {id},\lambda \) denoting the obvious inclusion. But this implies \( H\left( {\eta \lambda }\right) = {id} \) and so \( {\eta \lambda }\left( 1\right) = 1 \) . Since \( {\eta \lambda } \) is \( {\Lambda V} \) -linear, \( {\eta \lambda } = {id} \) . Now Hess’ theorem 29.9, asserts that \( \operatorname{cat}\left( {{\Lambda V}, d}\right) \leq m \) .
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Yes
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Theorem 29.16 \( \left\lbrack {55}\right\rbrack \; \) Let \( \left( {{\Lambda V}, d}\right) \) be a minimal Sullivan algebra such that \( V = \) \( {\left\{ {V}^{i}\right\} }_{i \geq 2} \) and has finite type. Then\n\n\[ \operatorname{cat}\left( {{\Lambda V}, d}\right) = \mathrm{e}\left( {\operatorname{Hom}\left( {{\Lambda V},\mathbb{k}}\right), d}\right) . \]
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proof of Theorem 29.16: Denote \( \left( {\operatorname{Hom}\left( {{\Lambda V},\mathbb{k}}\right), d}\right) \) by \( \left( {M, d}\right) \) . The finite type restriction implies that \( \left( {{\Lambda V}, d}\right) = \left( {\operatorname{Hom}\left( {M,\mathbb{R}}\right), d}\right) \) . Since \( \operatorname{cat}\left( {{\Lambda V}, d}\right) = \) \( \operatorname{mcat}\left( {{\Lambda V}, d}\right) \geq \mathrm{e}\left( {M, d}\right) - \) Proposition 29.15 - we have only to show that\n\n\[ \mathrm{e}\left( {M, d}\right) > \operatorname{mcat}\left( {{\Lambda V}, d}\right) . \]\n\nLet \( \varphi : \left( {P, d}\right) \overset{ \simeq }{ \rightarrow }\left( {M, d}\right) \) be a semifree resolution. Since \( \left( {\operatorname{Hom}\left( {M,\mathbb{k}}\right), d}\right) = \) \( \left( {{\Lambda V}, d}\right) \) this dualizes to a semifree resolution \( \operatorname{Hom}\left( {\varphi ,\mathbb{R}}\right) : \left( {{\Lambda V}, d}\right) \overset{ \simeq }{ \rightarrow }\operatorname{Hom}\left( {P,\mathbb{R}}\right) \) . Let \( z = \operatorname{Hom}\left( {\varphi ,\mathbb{k}}\right) 1 \in \operatorname{Hom}\left( {P,\mathbb{k}}\right) \).\n\nNow suppose \( \mathrm{e}\left( {M, d}\right) = r \) . Then the surjection \( \varrho : \left( {P, d}\right) \rightarrow \left( {P/{\Lambda }^{ > r}V \cdot P, d}\right) \) is injective in homology. It follows that the dual, \( \operatorname{Hom}\left( {\varrho ,\mathbb{R}}\right) \), induces a surjection\n\n\[ H\left( {\operatorname{Hom}\left( {P/{\Lambda }^{ > r}V \cdot P,\mathbb{k}}\right) }\right) \rightarrow H\left( {\operatorname{Hom}\left( {P,\mathbb{k}}\right) }\right) . \]\n\nIn particular there is a cocycle \( f \in \operatorname{Hom}\left( {P/{\Lambda }^{ > r}V \cdot P,\mathbb{R}}\right) \) such that \( \left\lbrack {f \circ \varrho }\right\rbrack = \left\lbrack z\right\rbrack \) .\n\nThe quasi-isomorphism \( {\Lambda V}\overset{ \simeq }{ \rightarrow }\operatorname{Hom}\left( {P,\mathbb{R}}\right) \) is given by \( \Phi \mapsto \Phi \cdot z \) . Thus it factors as\n\n\[ \left( {{\Lambda V}, d}\right) \overset{\alpha }{ \rightarrow }\left( {\operatorname{Hom}\left( {P/{\Lambda }^{ > r}V \cdot P,\mathbb{k}}\right), d}\right) \xrightarrow[]{\operatorname{Hom}\left( {\varrho ,\mathbb{k}}\right) }\left( {\operatorname{Hom}\left( {P,\mathbb{k}}\right), d}\right) \]\n\nwhere \( {\alpha \Phi } = \Phi \cdot f \) . Now for any \( g \in \operatorname{Hom}\left( {P/{\Lambda }^{ > r}V \cdot P,\mathbb{R}}\right),
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Yes
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Proposition 30.6 The fibration \( p : X \rightarrow Y \) (30.4) satisfies\n\n\[ \operatorname{cat}X \leq \left( {\operatorname{cat}Y + 1}\right) \left( {\operatorname{cat}F + 1}\right) - 1 \]\n\nand this inequality is best possible, even for rational spaces.
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proof: Let \( \operatorname{cat}Y = m \) so that \( Y \) is the union of \( m + 1 \) open sets \( {U}_{\alpha } \) each contractible in \( X \) . The inclusion \( {\lambda }_{\alpha } \) of \( {p}^{-1}\left( {U}_{\alpha }\right) \) in \( X \) is then homotopic to a map \( {p}^{-1}\left( {U}_{\alpha }\right) \rightarrow F\overset{j}{ \rightarrow }X \) and it follows that \( \operatorname{cat}{\lambda }_{\alpha } \leq \operatorname{cat}j \leq \operatorname{cat}F \) (Lemma 27.1). Thus \( {p}^{-1}\left( {U}_{\alpha }\right) \) is the union of \( \left( {n + 1}\right) \) open sets each contractible in \( X \), and so \( \operatorname{cat}X + 1 \leq \left( {\operatorname{cat}Y + 1}\right) \left( {\operatorname{cat}F + 1}\right) . \n\nTo see that this inequality can be sharp recall from Proposition 17.9 that spatial realization \( \mid \; \mid \) converts a relative Sullivan algebra to a Serre fibration. Thus in Example 1, \( \left| {\Lambda \left( {x, y, z, u, v}\right), d}\right| \rightarrow \left| {\Lambda \left( {x, y}\right), d}\right| \) is a Serre fibration with fibre \( \left| {\Lambda \left( {u, v}\right) ,\bar{d}}\right| \) . Denote \( \left| {\Lambda \left( {x, y}\right), d}\right| \) by \( Y \) and convert this to a fibration \( X \rightarrow Y \) with \( X \simeq \left| {\Lambda \left( {x, y, u, v}\right), d}\right| \) and fibre \( F \simeq \left| {\Lambda \left( {u, v}\right) ,\bar{d}}\right| \) . Now by the Corollary to Proposition 29.4 the LS category of the realization is the category of the model. Thus Example 1 translates to\n\n\[ \operatorname{cat}X = \left( {r + 1}\right) \left( {n + 1}\right) - 1 = \left( {\operatorname{cat}Y + 1}\right) \left( {\operatorname{cat}F + 1}\right) - 1. \]
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Yes
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Proposition 30.6 The fibration \( p : X \rightarrow Y \) (30.4) satisfies\n\n\[ \operatorname{cat}X \leq \left( {\operatorname{cat}Y + 1}\right) \left( {\operatorname{cat}F + 1}\right) - 1 \]\n\nand this inequality is best possible, even for rational spaces.
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proof: Let \( \operatorname{cat}Y = m \) so that \( Y \) is the union of \( m + 1 \) open sets \( {U}_{\alpha } \) each contractible in \( X \) . The inclusion \( {\lambda }_{\alpha } \) of \( {p}^{-1}\left( {U}_{\alpha }\right) \) in \( X \) is then homotopic to a map \( {p}^{-1}\left( {U}_{\alpha }\right) \rightarrow F\overset{j}{ \rightarrow }X \) and it follows that \( \operatorname{cat}{\lambda }_{\alpha } \leq \operatorname{cat}j \leq \operatorname{cat}F \) (Lemma 27.1). Thus \( {p}^{-1}\left( {U}_{\alpha }\right) \) is the union of \( \left( {n + 1}\right) \) open sets each contractible in \( X \), and so \( \operatorname{cat}X + 1 \leq \left( {\operatorname{cat}Y + 1}\right) \left( {\operatorname{cat}F + 1}\right) . \n\nTo see that this inequality can be sharp recall from Proposition 17.9 that spatial realization \( \mid \; \mid \) converts a relative Sullivan algebra to a Serre fibration. Thus in Example 1, \( \left| {\Lambda \left( {x, y, z, u, v}\right), d}\right| \rightarrow \left| {\Lambda \left( {x, y}\right), d}\right| \) is a Serre fibration with fibre \( \left| {\Lambda \left( {u, v}\right) ,\bar{d}}\right| \) . Denote \( \left| {\Lambda \left( {x, y}\right), d}\right| \) by \( Y \) and convert this to a fibration \( X \rightarrow Y \) with \( X \simeq \left| {\Lambda \left( {x, y, u, v}\right), d}\right| \) and fibre \( F \simeq \left| {\Lambda \left( {u, v}\right) ,\bar{d}}\right| \) . Now by the Corollary to Proposition 29.4 the LS category of the realization is the category of the model. Thus Example 1 translates to\n\n\[ \operatorname{cat}X = \left( {r + 1}\right) \left( {n + 1}\right) - 1 = \left( {\operatorname{cat}Y + 1}\right) \left( {\operatorname{cat}F + 1}\right) - 1. \]
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Yes
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Example 2 Fibrations with \( \operatorname{cat}X = 1 \) . We construct examples of fibrations with \( \operatorname{cat}X = {\operatorname{cat}}_{0}X = 1 \), and \[ {\operatorname{cat}}_{0}Y = n\;\text{ and }\;n \leq {\operatorname{cat}}_{0}F \leq n + 1, \] for any \( n \geq 1 \) . For this, let \( {p}_{i} : {S}_{i}^{7} \rightarrow {S}_{i}^{4} \) be a copy of the Hopf fibration, with fibre \( {S}_{i}^{3},1 \leq i \leq n \) . Convert the composite map \( \mathop{\bigvee }\limits_{{i = 1}}^{n}{S}_{i}^{7} \rightarrow \mathop{\prod }\limits_{{i = 1}}^{n}{S}_{i}^{7} \rightarrow \mathop{\prod }\limits_{{i = 1}}^{n}{S}_{i}^{4} \) into a fibration \( p : X \rightarrow Y \) with fibre \( F : Y = \mathop{\prod }\limits_{{i = 1}}^{n}{S}_{i}^{4} \) and \( X \simeq \mathop{\bigvee }\limits_{{i = 1}}^{n}{S}_{i}^{7} \) . In particular \( \operatorname{cat}X = {\operatorname{cat}}_{0}X = 1 \), and \( {\operatorname{cat}}_{0}Y = \operatorname{cat}Y = n \) .
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Now \( {\pi }_{ * }\left( {S}_{\alpha }^{4}\right) \otimes \mathbb{Q} = \mathbb{Q}{e}_{\alpha } \oplus \mathbb{Q}{e}_{\alpha }^{\prime } \) with \( \deg {e}_{\alpha } = 4 \) and \( \deg {e}_{\alpha }^{\prime } = 7 \) (Example 1, \( §{15}\left( \mathrm{\;d}\right) ) \) and \( {e}_{\alpha }^{\prime } \) is in the image of \( {\pi }_{ * }\left( {p}_{\alpha }\right) \otimes \mathbb{Q} \) by construction. Thus each \( {e}_{\alpha }^{\prime } \in \) \( \operatorname{Im}{\pi }_{ * }\left( p\right) \otimes \mathbb{Q} \) . On the other hand \( X \) is 6-connected and so \( {\partial }_{ * } \otimes \mathbb{Q} \) is a injective in \( {\pi }_{4}\left( Y\right) \otimes \mathbb{Q} \) . It follows that \( \ker \left( {{\partial }_{ * } \otimes \mathbb{Q}}\right) = {\pi }_{7}\left( Y\right) \otimes \mathbb{Q} \) and Proposition 30.9(ii) asserts that \( n \leq {\operatorname{cat}}_{0}F \leq n + 1 \) .
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No
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Proposition 30.10 If the fibration \( p : X \rightarrow Y \) has cross-section then \( {\operatorname{cat}}_{0}X \geq \max \left( {{\operatorname{cat}}_{0}Y,{\operatorname{cat}}_{0}F}\right) = {\operatorname{cat}}_{0}\left( {Y \vee F}\right) .
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proof: Recall from the Example in \( §{28} \) (a) that \( \max \left( {{\operatorname{cat}}_{0}Y,{\operatorname{cat}}_{0}F}\right) = {\operatorname{cat}}_{0}(Y \vee \) \( F) \) . Let \( s : Y \rightarrow X \) be the cross-section: \( {ps} = i{d}_{Y} \) . Thus exhibits \( Y \) as a retract of \( X \) so \( {\operatorname{cat}}_{0}Y \leq {\operatorname{cat}}_{0}X \) and it exhibits \( {\pi }_{ * }\left( p\right) \) as surjective, so the inclusion \( j : F \rightarrow Y \) induces an injection \( {\pi }_{ * }\left( j\right) \otimes \mathbb{Q} \) of rational homotopy groups. By the Mapping theorem 28.6, \( {\operatorname{cat}}_{0}F \leq {\operatorname{cat}}_{0}X \) as well.
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Yes
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Example 3 Fibrations with \( {\operatorname{cat}}_{0}X = \max \left( {{\operatorname{cat}}_{0}Y,{\operatorname{cat}}_{0}F}\right) \) .
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For any simply connected topological spaces \( Y \) and \( Z \) convert the map \( \left( {i{d}_{Y}\text{, const.}}\right) : Y \vee Z \rightarrow Y \) into a fibration \( p : X \rightarrow Y \) with \( X \simeq Y \vee Z \) , as described in \( §2\left( \mathrm{c}\right) \) . The inclusion \( Y \rightarrow Y \vee Z \) then defines a cross-section \( s \) of this fibration; in particular the fibre \( F \) is simply connected.\n\nMoreover, the inclusion \( Z \rightarrow Y \vee Z \) defines an inclusion \( i : Z \rightarrow F \) . Let \( j : F \rightarrow X \) be the inclusion. Since the fibration has a cross-section, \( {\pi }_{ * }\left( j\right) \otimes \mathbb{Q} \) is injective. Since \( {ji} \) is homotopic to the inclusion \( Z \rightarrow Y \vee Z,{\pi }_{ * }\left( {ji}\right) \otimes \mathbb{Q} \) is injective. Hence \( {\pi }_{ * }\left( i\right) \otimes \mathbb{Q} \) is also injective and the Mapping theorem 28.6 asserts that\n\n\[ \n{\operatorname{cat}}_{0}Z \leq {\operatorname{cat}}_{0}F \leq {\operatorname{cat}}_{0}X \n\]\n\nThus \( {\operatorname{cat}}_{0}X = \max \left( {{\operatorname{cat}}_{0}Y,{\operatorname{cat}}_{0}Z}\right) = \max \left( {{\operatorname{cat}}_{0}Y,{\operatorname{cat}}_{0}F}\right) \) .
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Yes
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Proposition 31.6 Let \( p : X \rightarrow Y \) be a Serre fibration with fibre \( F \), and suppose \( Y \) is simply connected with rational homology of finite type. If \( {H}^{ * }\left( {X;\mathbb{k}}\right) \rightarrow \) \( {H}^{ * }\left( {F;\mathbf{k}}\right) \) is surjective then the holonomy representation of \( {L}_{Y} \) in \( {H}_{ * }\left( {F;\mathbf{k}}\right) \) is trivial.
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proof: Let \( \left( {{\Lambda V} \otimes {\Lambda W}, d}\right) \) be a minimal Sullivan model for the fibration as in (15.4). Then for any cohomology class \( \alpha \in H\left( {{\Lambda W},\bar{d}}\right) \) there is a \( d \) -cocycle \( \Psi \in {\Lambda V} \otimes {\Lambda W} \) of the form \( \Psi = 1 \otimes {\Psi }_{0} + {\Psi }_{1} + \cdots \) such that \( {\Psi }_{i} \in {\Lambda }^{i}V \otimes {\Lambda W} \) and \( {\Psi }_{0} \) is a \( \bar{d} \) -cocycle representing \( \alpha \) .\n\nAs in the introduction, choose a basis \( {v}_{i} \) of \( V \) and define derivations \( {\theta }_{i} \) of \( \left( {{\Lambda W},\bar{d}}\right) \) by \( d\left( {1 \otimes \Phi }\right) - 1 \otimes \bar{d}\Phi - \sum {v}_{i} \otimes {\theta }_{i}\Phi \in {\Lambda }^{ \geq 2}V \otimes {\Lambda W} \) . Then (formula (31.1)) the holonomy representation in \( H\left( {{\Lambda W},\bar{d}}\right) \) is given by \( \alpha \cdot {x}_{i} = \pm H\left( {\theta }_{i}\right) \alpha \), where \( {x}_{i} \) is the dual basis of the homotopy Lie algebra \( L \) of \( \left( {{\Lambda V}, d}\right) \) . Here we have \( {d\Psi } = 0 \) and so \( \sum {v}_{i} \otimes {\theta }_{i}{\Psi }_{0} = - \left( {{id} \otimes \bar{d}}\right) {\Psi }_{1} \) . Thus each \( H\left( {\theta }_{i}\right) = 0 \) and the holonomy representation in \( {H}_{ * }\left( {F;\mathbb{k}}\right) \) is trivial (Theorem 31.3).
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Yes
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Lemma 31.7 Suppose \( \sigma : M \rightarrow M \) is a linear map of non-zero degree in a graded vector space \( M = {\left\{ {M}^{p}\right\} }_{p \in \mathbb{Z}} \) of finite type. Then the following conditions are equivalent:\n\n(i) The dual linear map in \( \operatorname{Hom}\left( {M,\mathbb{k}}\right) \) is locally nilpotent.\n\n(ii) For each \( p \) there is an integer \( k\left( p\right) \) such that \( {M}^{p} \cap \operatorname{Im}{\sigma }^{k\left( p\right) } = 0 \) .\n\n(iii) If \( {\left( {w}_{k}\right) }_{k \geq 0} \) is an infinite sequence of elements of \( M \) such that \( \sigma {w}_{k + 1} = {w}_{k} \) , \( k \geq 0 \) then \( {w}_{k} = 0, k \geq 0 \) .\n\n(iv) \( \mathop{\bigcap }\limits_{{k = 0}}^{\infty }\operatorname{Im}{\sigma }^{k} = 0 \) .
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proof of Lemma 31.7: \( \;\left( i\right) \Leftrightarrow \left( {ii}\right) : \) Denote the linear map dual to \( \sigma \) by \( f \) . Then\n\n\[ \left\langle {M,{f}^{k}\left( {\operatorname{Hom}{\left( M,\mathbb{k}\right) }_{p}}\right) }\right\rangle = \left\langle {{M}^{p} \cap \operatorname{Im}{\sigma }^{k},\operatorname{Hom}{\left( M,\mathbb{k}\right) }_{p}}\right\rangle . \]\n\nThus if (ii) holds then \( {f}^{k\left( p\right) } = 0 \) in \( \operatorname{Hom}{\left( M,\mathbb{R}\right) }_{p}, p \in \mathbb{Z} \), and \( f \) is locally nilpotent. Conversely, if (i) holds then \( {f}^{k\left( p\right) } = 0 \) in \( \operatorname{Hom}{\left( M,\mathbb{R}\right) }_{p} \) for some \( k\left( p\right) \), because this vector space is finite dimensional. Thus \( {M}^{p} \cap \operatorname{Im}{\sigma }^{k\left( p\right) } = 0 \) and (ii) holds.\n\n(ii) \( \Leftrightarrow \) (iii): Clearly (ii) \( \Rightarrow \) (iii). If (ii) fails then for some \( p \) and each \( k,{M}^{p} \cap \operatorname{Im}{\sigma }^{k} \neq 0 \) . Set \( {A}_{k} = {\left( {\sigma }^{k + 1}\right) }^{-1}\left( {{M}^{p}-\{ 0\} }\right) \) . Then \( {A}_{0}\overset{\sigma }{ \leftarrow }{A}_{1}\overset{\sigma }{ \leftarrow } {A}_{2}\overset{\sigma }{ \leftarrow }\cdots \) is an infinite sequence of linear maps between non-void affine spaces. For dimension reasons the sequences \( {A}_{k} \supset \sigma \left( {A}_{k + 1}\right) \supset {\sigma }^{2}\left( {A}_{k + 2}\right) \supset \cdots \) must stabilize at some integer \( K\left( k\right) : {\sigma }^{n}\left( {A}_{k + n}\right) = {\sigma }^{K\left( k\right) }\left( {A}_{k + K\left( k\right) }\right) \) for \( n \geq K\left( k\right) \).\n\nPut \( {E}_{k} = {\sigma }^{K\left( k\right) }\left( {A}_{k + K\left( k\right) }\right) \) . Then for \( n \) large enough\n\n\[ \sigma \left( {E}_{k + 1}\right) = \sigma \left( {{\sigma }^{n}\left( {A}_{k + 1 + n}\right) }\right) = {\sigma }^{n + 1}\left( {A}_{k + n + 1}\right) = {E}_{k}. \]\n\nThus we may inductively construct elements \( {w}_{k} \in {E}_{k} \) so that \( \sigma {w}_{k + 1} = {w}_{k} \) . Moreover, since \( {A}_{k} = {\left( {\sigma }^{k + 1}\right) }^{-1}\left( {{M}^{p}-\{ 0\} }\right) \) no element in \( {A}_{k} \) is zero, so \( {w}_{k} \neq 0 \) , \( k \geq 0 \).\n\n(ii) \( \Leftrightarrow \) (iv): Clearly (ii) \( \Rightarrow \) (iv) and the reverse implication follows\n\nfrom decreasing sequence \( {M}^{p} \supset {M}^{p} \cap \operatorname{Im}\sigma \supset {M}^{p} \cap \operatorname{Im}{\sigma }^{2} \supset \cdots \) and the fact that \( {M}^{p} \) is finite dimensional.
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Yes
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