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Example 2 Contractible Sullivan algebras have contractible realizations. | 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 ... | No |
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( \... | 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 \(... | Yes |
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 pa... | 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 Proposi... | Yes |
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 | 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 \( {\varepsil... | Yes |
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 ,{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 \( {t}_{Y}f \sim \left| {{S}_{ * }\left( f\right) }\right| {t}_{X} \) .... | Yes |
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. | 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 \( ... | Yes |
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 ... | 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}}\righ... | Yes |
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 \( H\\left( M\\right) \) . | 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} \\right... | Yes |
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 isomo... | 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}_{\inft... | Yes |
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 \( \psi \) . | 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... | Yes |
Proposition 19.1 With the notation above, \( \varrho \) is a quasi-isomorphism. | 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\lb... | Yes |
Lemma 20.1 Every A-module \( M \) has a free resolution. | 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 \( A \) -linea... | No |
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) ... | 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... | Yes |
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\} . | 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 \( {\operatorname{Ext}}_{A}^{n}\left( {M, - }\right) = 0, n > k \) . He... | Yes |
Proposition 20.10 For any A-module, \( M \) , \[ {\operatorname{projdim}}_{A}\left( M\right) = \sup \left\{ {k \mid {\operatorname{Ext}}_{A}^{k}\left( {M, - }\right) \text{ is non-zero }}\right\} . \] | 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 \( {\operatorname{Ext}}_{A}^{n}\left( {M, - }\right) = 0, n > k \) . He... | Yes |
Example 1 The Eilenberg-Moore spectral sequence. | 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 ... | Yes |
If \( L \) is a graded Lie algebra then a representation ad : \( L \rightarrow \operatorname{Hom}\left( {L, L}\right) \) is defined by\n\n\[ \left( {\operatorname{ad}x}\right) \left( y\right) = \left\lbrack {x, y}\right\rbrack ,\;x, y \in L. \] | This is called the adjoint representation, and (by the Jacobi identity) makes \( L \) into an \( L \) -module. The submodules are precisely the ideals in \( L \) . | No |
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 }\righ... | In particular for \( x \in E, y \in L \) we have \( \left\lbrack {x, y}\right\rbrack = 0 \) in \( E \oplus L \) . | Yes |
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 }... | 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 ... | No |
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 ... | 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 \( {UL}\left( k\right) \) . Mo... | Yes |
Proposition 21.3 The inclusion \( L \rightarrow {UL} \) is an isomorphism of \( L \) onto the graded Lie algebra of primitive elements in UL. | 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... | Yes |
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. | 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) \\), satisfie... | Yes |
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|... | 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 }}^{... | Yes |
Lemma 22.2 \( {\theta }_{g} \) is a coderivation in \( {\Lambda V} \) . It is the unique coderivation that extends \( g \) and decreases wordlength by \( k - 1 \) . | proof: The coderivation property is a simple calculation. Uniqueness is proved in the same way as in Lemma 22.1. | No |
Proposition 22.3 The inclusion \( \mathbf{k} \rightarrow \left( {{\Lambda sL} \otimes {UL}, d}\right) \) is a quasi-isomorphism. | 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 ... | Yes |
Proposition 22.4 Right multiplication makes \( \\left( {{\\Lambda sL} \\otimes {UL}, d}\\right) \) into a right semifree \( \\left( {{UL}, d}\\right) \) -module. | 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 in... | Yes |
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| \c... | 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 1... | Yes |
Proposition 22.8 The linear map \( \varrho \) is a natural quasi-isomorphism of complexes,\n\n\[ \left( {{C}_{ * }\left( {\mathbb{L}}_{V}\right), d}\right) \overset{ \simeq }{ \rightarrow }\left( {{sV} \oplus \mathbb{k},\bar{d}}\right) \] | 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... | Yes |
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. | 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. | Yes |
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)... | 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 \( \Phi \in {\Lambda }^{p}{\left( sL\right) }^{\sharp } \)... | Yes |
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} + ... | 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 \( {d}_{L} : L \rightarrow L \) an... | Yes |
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} \... | 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\... | No |
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 coal... | 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 differen... | Yes |
Example 4 Sullivan algebras \( \left( {{\Lambda W}, d}\right) \) for which \( {H}^{2k}\left( {{\Lambda W}, d}\right) = 0,1 \leq k \leq n \) . | 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 line... | Yes |
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\[ H\left( L\right) \overset{ \cong }{ \rightarrow }E \] | 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{ \con... | Yes |
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 | \[ {E}_{2} = {\operatorname{Ext}}_{{UH}\left( L\right) }\left( {\mathbb{k},\mathbb{k}}\right) \Rightarrow H{C}^{ * }\left( {L,{d}_{L}}\right) . \] | Yes |
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) \\con... | (This follows, respectively, from Corollary 10.10, Theorem 21.5 and Proposition 21.6.) | No |
The free Lie model of a sphere. | 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... | Yes |
Proposition 24.4 With the hypotheses above the diagram\n\n\n\ncommutes, and identifies \( {sH}\left( \eta \right) \) with the Hurewicz homomorphism hur \( x \) . | 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\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... | Yes |
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 \( f \) . | 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... | Yes |
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... | 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}_{\... | Yes |
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}_{ * }\le... | 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 \( {EZ} \) . | No |
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 | \[ \Theta : U\left( {{\mathbb{L}}_{V}, d}\right) \overset{ \simeq }{ \rightarrow }{C}_{ * }\left( {\Omega X}\right) \] 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)... | Yes |
Proposition 26.6 The map \( {\int }_{ * } : {C}_{ * }\langle \Gamma \rangle \rightarrow \left( {U{\mathbb{L}}_{V}, d}\right) \) is a chain algebra quasi-isomorphism. | 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| ... | Yes |
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. | 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 \( {\Del... | Yes |
Lemma 27.1 Suppose \( f : X \rightarrow Y \) is a continuous map.\n\n(i) If \( {f}^{\prime } \sim f \) then \( \operatorname{cat}{f}^{\prime } = \operatorname{cat}f \) . | 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 hom... | No |
Lemma 27.3 For any continuous map \( g : A \rightarrow X \) , | \[ \operatorname{cat}\left( {X{ \cup }_{g}{CA}}\right) \leq \operatorname{cat}X + 1. \] 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... | Yes |
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 \... | 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}_... | Yes |
Proposition 27.5 Suppose \( X \) is an \( \left( {r - 1}\right) \) -connected CW complex of dimension \( d \) (some \( r \geq 1 \) ). Then\n\n\[ \operatorname{cat}X \leq d/r \] | 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 h... | Yes |
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... | 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}_... | Yes |
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 \( Y \) . | ## 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 ... | Yes |
Proposition 27.13 If \( B \) is normal then\n\n\[ \operatorname{cl}\left( {E{ \cup }_{j}{CF}}\right) \leq \operatorname{cl}B\;\text{ and }\;\operatorname{cat}\left( {E{ \cup }_{j}{CF}}\right) \leq \operatorname{cat}B. \] | 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... | Yes |
The LS category of the Ganea spaces \( {P}_{n}X \) . | 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 t... | Yes |
Proposition 27.14 If \( X \) is a path connected normal space then for any coefficient ring \( \mathbb{k} \) , \[ \mathrm{c}\left( {X;\mathbb{k}}\right) \leq \mathrm{e}\left( {X;\mathbb{k}}\right) \leq \operatorname{cat}X \leq \operatorname{cl}X. \] | 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 \( {\... | Yes |
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. | 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 \( m \) -con... | Yes |
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 \( \mathrm{e}\left( {Y;\mathbf{k}}\right) = \mathrm{e}\left( {X;\mathbf{k}}\right) \) . | 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 ... | No |
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 \( {P}_{m}f \) . | 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 ... | Yes |
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) \( {e}_{0}X = e\left( {X;\mathbb{Q}}\right) = e\left( {{X}_{\mathbb{Q}};\mathbb{Q}}\right) \) . | 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 \( ... | Yes |
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 \( \operatorname{cat}X \leq \operatorname{cat}Y \) . | 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 equi... | Yes |
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, \( P \) . | 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 \( §{27}\left( \mathrm{\;d}\right) \) . | No |
Theorem 28.5 If \( X \) is a simply connected topological space then\n\n(i) \( {e}_{0}X \leq {\operatorname{cat}}_{0}X \leq {\operatorname{cl}}_{0}X \leq {\operatorname{cat}}_{0}X + 1 \) . | 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}... | Yes |
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\[{\operatorname{cat}}_{0}X \leq {\operatorname{cat}}_{0}Y\] | 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 \( X{ \times }_{Y}{PY} \) in... | Yes |
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 b... | \[ \rightarrow {X}^{n + 1}\overset{{g}_{n}}{ \rightarrow }\cdots \rightarrow {X}^{3}\overset{{g}_{2}}{ \rightarrow }{X}^{2} = X \] 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 f... | Yes |
Free loop spaces have infinite rational category. | 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}^... | Yes |
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 \[ \left\lbrack {\alpha ,\beta }\right\rbrack = 0,\;\beta \in {L}_{X}. \] | 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... | Yes |
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) \( \dim {G}_{ * }^{\mathbb{Q}}\left( X\right) \leq {\operatorname{cat}}_{0}X \) . | 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 \( \left( {{2k} - 1}\right) \) -conne... | Yes |
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) \( \dim {G}_{ * }^{\mathbb{Q}}\left( X\right) \leq {\operatorname{cat}}_{0}X \) . | 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 \( \left( {{2k} - 1}\right) \) -conne... | Yes |
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... | 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 \... | Yes |
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\[ \operatorname{cat}\left( {{\Lambda V}, d}\right) \leq n/r \] | 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. | Yes |
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\[ \operatorname{cat}\left( {{\Lambda V}, d}\right) = {\operatorname{cat}}_{0}X\;\text{ and }\;e\left( {{\Lambda V}, d}\right) = {e}_{0}X. \] | 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 ... | Yes |
Example 1 A space \( X \) satisfying \( {c}_{0}X < {e}_{0}X \) . | 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\... | Yes |
A space \( X \) satisfying \( {e}_{0}X < {\operatorname{cat}}_{0}X \) . | 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\nThu... | Yes |
Let \( X \) be a simply connected topological space with rational homology of finite type. If \( X \) is formal then | \[ {c}_{0}\left( X\right) = {e}_{0}\left( X\right) = {\operatorname{cat}}_{0}X = {\operatorname{cl}}_{0}X. \] 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 Theo... | Yes |
We show for coformal spaces that\n\n\[ \n{e}_{0}\left( X\right) = {\operatorname{cat}}_{0}\left( X\right) = {\operatorname{cl}}_{0}X \n\] | 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( ... | No |
Example 6 Minimal Sullivan algebras \( \left( {{\Lambda V}, d}\right) \) with \( V = {V}^{\text{odd }} \) and \( \dim V < \infty \) . | 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 = \... | Yes |
Example 7 \( \left( {{\Lambda V}, d}\right) = \Lambda \left( {a, b, x, y, z}\right) \) with \( {dx} = {a}^{2},{dy} = {b}^{2} \) and \( {dz} = {ab} \) . | 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 c... | Yes |
Example 1 \( \;\mathrm{e}\left( {\left( {{\Lambda V}, d}\right) \otimes \left( {{\Lambda W}, d}\right) }\right) = \mathrm{e}\left( {{\Lambda V}, d}\right) + \mathrm{e}\left( {{\Lambda W}, d}\right) \) . | 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 i... | Yes |
Example 2 \( {\operatorname{cat}}_{0}X - {\mathrm{e}}_{0}X \) can be arbitrarily large. | 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 \( \Lambda \left( {x, y, z, t, u, v,\ldots }\right) \) wit... | Yes |
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 \( \\le... | 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 \( \\... | Yes |
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 }^{... | 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... | Yes |
Field extension preserves category. | 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 tha... | Yes |
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}}_{... | 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}\ri... | Yes |
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 \op... | 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... | Yes |
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( {{\La... | 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}... | Yes |
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. | 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... | Yes |
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. | 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... | Yes |
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... | 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 o... | No |
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) . | 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 \( {\operatornam... | Yes |
Example 3 Fibrations with \( {\operatorname{cat}}_{0}X = \max \left( {{\operatorname{cat}}_{0}Y,{\operatorname{cat}}_{0}F}\right) \) . | 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 \( Y \rightarrow Y \vee Z \) t... | Yes |
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 ... | 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 {... | Yes |
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 locall... | 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},\oper... | Yes |
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