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Lemma 4.12 For \( n \geq 0,{H}_{ * }\left( {{\Delta }^{n},\partial {\bigtriangleup }^{n};\mathbf{k}}\right) \) is a free module concentrated in degree \( n \) with single basis element \( \left\lbrack {\Delta }^{n}\right\rbrack \) | \[ {H}_{ * }\left( {{\Delta }^{n},\partial {\Delta }^{n};\mathbb{k}}\right) = \mathbb{k} \cdot \left\lbrack {\Delta }^{n}\right\rbrack \] proof: We may suppose \( n > 0 \) . Regard \( {\Delta }^{n - 1} = \operatorname{Im}\left\langle {{e}_{0},\ldots ,{e}_{n - 1}}\right\rangle \) as one of the faces of \( {\Delta }^{n} ... | Yes |
Theorem 4.15 If \( \varphi : X \rightarrow Y \) is a weak homotopy equivalence then \( {C}_{ * }\left( \varphi \right) \) : \( {C}_{ * }\left( {X;\mathbb{R}}\right) \rightarrow {C}_{ * }\left( {Y;\mathbb{R}}\right) \) is a quasi-isomorphism. | proof: Recall the face maps \( {\lambda }_{i} = \left\langle {{e}_{0}\ldots {\widehat{e}}_{i}\ldots {e}_{n}}\right\rangle : {\Delta }^{n - 1} \rightarrow {\Delta }^{n} \) . We first observe that for each \( n \geq 0 \) and each \( \sigma : {\bigtriangleup }^{n} \rightarrow Y \) we can associate \( {\sigma }^{\prime } :... | Yes |
Theorem 4.18 (Cellular chain models) Every relative CW complex \( \left( {X, A}\right) \) has a cellular chain model \( m : \left( {{C}_{ * },\partial }\right) \rightarrow {C}_{ * }\left( {X, A;\mathbb{k}}\right) \), and \( m \) and each \( m\left( n\right) \) are always quasi-isomorphisms. | proof: We construct the morphisms \( m\left( n\right) \) inductively and observe in passing that they are quasi-isomorphisms.\n\nSuppose by induction that \( m\left( n\right) \) is constructed; we extend it to \( m\left( {n + 1}\right) \) as follows. Represent a basis element \( {c}_{\alpha } \in {C}_{n + 1} \) by a cy... | Yes |
Proposition 4.21 Suppose \( \pi \) is an abelian group. Then there exists a \( K\left( {\pi, n}\right) \) and any two have the same weak homotopy type. | proof: Suppose \( n \geq 2 \) . Write \( \pi \) as the quotient of a free abelian group on generators \( {g}_{\alpha } \) divided by relations \( {r}_{\beta } \) . The proof of the Hurewicz theorem (4.19) identifies \( {\pi }_{n}\left( {{ \vee }_{\alpha }{S}_{\alpha }^{n}}\right) = {\bigoplus }_{\alpha }\mathbb{Z}{g}_{... | Yes |
Proposition 5.3 (i) If \( \mathbb{k} \) is a field then \( \alpha \) is always an isomorphism and so \( \langle \) , \( \bar{)\text{ }{is}\text{ }{non}\text{-}{degenerate}\text{. }{In}\text{ }{particular}\text{,}{H}^{ * }\left( {X, A;{Ik}}\right) \text{ }{hasfinite}\text{ }{type}\text{ }{if}\text{ }{and}\text{ }{only}\... | proof: (i) This is just the assertion of Proposition 3.3 that \ | No |
Lemma 6.2 If \( \left( {R, d}\right) \rightarrow \left( {S, d}\right) \) is a morphism of dga’s, and if \( \left( {M, d}\right) \) is a semifree \( \left( {R, d}\right) \) -module then \( \left( {S{ \otimes }_{R}M, d}\right) \) is \( \left( {S, d}\right) \) -semifree. | proof: Let \( \{ M\left( k\right) \} \) be a semifree filtration for \( \left( {M, d}\right) \) . From the formula in Remark 6.1 we deduce\n\n\[ S{ \otimes }_{R}M\left( k\right) = S{ \otimes }_{R}M\left( {k - 1}\right) \oplus \left( {S \otimes Z\left( k\right) }\right), d : Z\left( k\right) \rightarrow S{ \otimes }_{R}... | Yes |
Lemma 6.3 Suppose an \( \\left( {R, d}\\right) \) -module \( \\left( {M, d}\\right) \) is the union of an increasing sequence \( M\\left( 0\\right) \\subset M\\left( 1\\right) \\subset \\cdots \) of submodules such that \( M\\left( 0\\right) \) and each \( M\\left( k\\right) /M(k - \\) 1) is \( \\left( {R, d}\\right) \... | proof: Put \( M\\left( {-1}\\right) = 0 \) . In the same way as in the Remark we may write\n\n\\[ \nM\\left( k\\right) = M\\left( {k - 1}\\right) \\oplus \\left( {R \\otimes \\left\\lbrack {{\\bigoplus }_{\\ell = 0}^{\\infty }Z\\left( {k,\\ell }\\right) }\\right\\rbrack }\\right) , \n\\]\n\nwith \( Z\\left( {k,\\ell }\... | Yes |
Proposition 6.4 Suppose \( \left( {M, d}\right) \) is semifree and \( \eta \) is a quasi-isomorphism. Then\n\n(i) \( {\operatorname{Hom}}_{R}\left( {M,\eta }\right) \) is a quasi-isomorphism. | proof: (i) As remarked in Lemma 3.2 it is sufficient to show that given \( f \in \) \( {\operatorname{Hom}}_{R}\left( {M, P}\right) \) and \( g \in {\operatorname{Hom}}_{R}\left( {M, Q}\right) \) satisfying \( d\left( f\right) = 0 \) and \( \eta \circ f = d\left( g\right) \) we can find \( {f}^{\prime } \in {\operatorn... | Yes |
Proposition 6.7 Suppose \( \left( {M, d}\right) \) and \( \left( {{M}^{\prime }, d}\right) \) are \( \left( {R, d}\right) \) -semifree.\n\n(i) If \( f \) and \( g \) are quasi-isomorphisms then so is \( {\operatorname{Hom}}_{R}\left( {f, g}\right) \) . | proof: (i) Since \( \left( {M, d}\right) \) and \( \left( {{M}^{\prime }, d}\right) \) are semifree, Proposition 6.4 (iii) asserts that \( f \) is an equivalence. Thus there is an inverse equivalence \( {f}^{\prime } : \left( {{M}^{\prime }, d}\right) \rightarrow \) \( \left( {M, d}\right) \) and \( R \) -linear maps \... | Yes |
Theorem 6.10 Suppose \( \left( {M, d}\right) \) is \( \left( {R, d}\right) \) -semifree and \( \left( {{M}^{\prime }, d}\right) \) is \( \left( {S, d}\right) \) - semifree.\n\n(i) If \( \varphi, f \) and \( g \) are quasi-isomorphisms, so is \( {\operatorname{Hom}}_{\varphi }\left( {f, g}\right) \) .\n\n(ii) If \( \var... | proof: As observed in Lemma 6.2, \( \left( {S{ \otimes }_{R}M, d}\right) \) is \( \left( {S, d}\right) \) -semifree. Moreover, since \( \left( {M, d}\right) \) is \( \left( {R, d}\right) \) -semifree\n\n\[ \varphi { \otimes }_{R}{id} : \left( {R{ \otimes }_{R}M, d}\right) \rightarrow \left( {S{ \otimes }_{R}M, d}\right... | Yes |
Theorem 6.12 With the hypotheses above:\n\n\\( f \\) is a quasi-isomorphism \\( \\Leftrightarrow f{ \\otimes }_{\\varphi }{H}_{0}\\left( \\varphi \\right) \\) is a quasi-isomorphism. | proof: We have only to prove \\( \\Leftarrow \\) since the reverse implication is just Remark 2. Define a decreasing sequence of differential ideals\n\n\\[ \nR \\supset {I}^{1} \\supset {J}^{1} \\supset {I}^{2} \\supset \\cdots \\supset {I}^{n} \\supset {J}^{n} \\supset \\cdots \n\\]\n\nby setting\n\n\\[ \n{\\left( {I}... | Yes |
Theorem 7.10 Suppose \( \mathbb{k} \) is a field, \( \pi : X \rightarrow Y \) is a fibration with \( Y \) simply connected and that one of the graded \( \mathbb{k} \) -vector spaces \( {H}^{ * }\left( Y\right) ,{H}^{ * }\left( F\right) \) has finite type. If \( \mathcal{D} \) is a square weakly equivalent to \( {\mathc... | proof: Suppose given a quasi-isomorphism \( \mathcal{D} \rightarrow \mathcal{D}\left( 1\right) \) as above. Then we have the commutative diagram\n\n\n\nwhich identifies \( m \) as a quasi-isomorphism. From this we de... | Yes |
Theorem 8.5 With the notation above, assume \( G \) is path connected and \( {C}_{ * }\left( \alpha \right) \) is a quasi-isomorphism. Then\n\n\[ \n{C}_{ * }\left( \varphi \right) \;\text{ is a quasi-isomorphism }\; \Leftrightarrow \;{C}_{ * }\left( \psi \right) \;\text{ is a quasi-isomorphism. }\n\] | proof: The proof of Proposition 6.6 shows that \( {C}_{ * }\left( P\right) \) and \( {C}_{ * }\left( {P}^{\prime }\right) \) admit \( {C}_{ * }\left( G\right) \) and \( {C}_{ * }\left( {G}^{\prime }\right) \) semifree resolutions\n\n\[ \nm : \left( {M, d}\right) \overset{ \simeq }{ \rightarrow }{C}_{ * }\left( P\right)... | Yes |
Lemma 8.7 Suppose \( \mathbf{k} \subset \mathbb{Q}, X \) is an \( \left( {r - 1}\right) \) -connected space and either \( r \geq 2 \) or \( r = 1 \) and \( {\pi }_{1}\left( X\right) \) is abelian. Then the Hurewicz homomorphism defines an isomorphism\n\n\[ \n{\pi }_{r}\left( X\right) { \otimes }_{\mathbb{Z}}\mathbb{k} ... | proof: By Theorem 4.19 the Hurewicz homomorphism is an isomorphism \( {\pi }_{r}\left( X\right) \) \( \overset{ \cong }{ \rightarrow }{H}_{r}\left( {X;\mathbb{Z}}\right) \) . Thus it defines an isomorphism \( {\pi }_{r}\left( X\right) { \otimes }_{\mathbb{Z}}\mathbb{k}\overset{ \cong }{ \rightarrow }{H}_{r}\left( {X;\m... | Yes |
Lemma 9.2 Let \( {\mathbb{F}}_{p} \) be the prime field of characteristic \( p \) . Then for all pairs of spaces \( \left( {X, A}\right) \) :\n\n\[ \n{H}_{ * }\left( {X, A;\mathbb{Z}}\right) \text{ is }\mathcal{P} - \text{ local }\; \Leftrightarrow \;{H}_{ * }\left( {X, A;{\mathbb{F}}_{p}}\right) = 0, p \notin \mathcal... | proof: Consider the long exact homology sequence associated with the short exact sequence\n\n\[ \n0 \rightarrow {C}_{ * }\left( {X, A;\mathbb{Z}}\right) \overset{\times p}{ \rightarrow }{C}_{ * }\left( {X, A;\mathbb{Z}}\right) \rightarrow {C}_{ * }\left( {X, A;{\mathbb{F}}_{p}}\right) \rightarrow 0.\n\]\n\nIt shows tha... | Yes |
Lemma 9.4 If \( X \) is an Eilenberg-MacLane space of type \( \left( {\pi, n}\right), n \geq 1 \), then\n\n\[ \pi \text{ is }\mathcal{P}\text{-local }\; \Rightarrow \;{H}_{ * }\left( {X;{\mathbb{F}}_{p}}\right) = {H}_{ * }\left( {{pt};{\mathbb{F}}_{p}}\right), p \notin \mathcal{P}. \] | proof: Suppose first that \( n = 1 \) . Reduce to the case \( \pi \) is finitely generated exactly as in Case 3 in the proof of Theorem 8.6. Since \( \mathbb{R} \) is a principal ideal domain, finitely generated \( \mathbb{k} \) -modules are the finite direct sums of cyclic \( \mathbb{k} \) - modules. Hence in this cas... | Yes |
Theorem 9.6 A continuous map \( \varphi : X \rightarrow Y \) between simply connected spaces is a \( \mathcal{P} \) -localization if and only if \( Y \) is \( \mathcal{P} \) -local and \( {H}_{ * }\left( {\varphi ;\mathbb{k}}\right) \) is an isomorphism. | proof: Since \( Y \) is \( \mathcal{P} \) -local, \( {\pi }_{ * }\left( Y\right) = {\pi }_{ * }\left( Y\right) { \otimes }_{\mathbb{Z}}\mathbb{k} \) and the homomorphism \( {\pi }_{ * }\left( X\right) { \otimes }_{\mathbb{Z}}\mathbb{k} \rightarrow {\pi }_{ * }\left( Y\right) \) is just \( {\pi }_{ * }\left( \varphi \ri... | No |
Simply connected spaces \( X \) and \( Y \) have the same rational homotopy type if and only if there is a chain of rational homotopy equivalences\n\n\[ X \leftarrow Z\left( 0\right) \rightarrow \cdots \leftarrow Z\left( k\right) \rightarrow Y. \] | The first assertion is immediate from Theorem 9.7. The second follows immediately from the Whitehead lifting lemma 1.5, because \( \left( {{X}_{\mathbb{Q}}, X}\right) \) and \( \left( {{Y}_{\mathbb{Q}}, Y}\right) \) are relative CW complexes. | No |
Lemma 10.3 If \( K \) is any simplicial set then any \( \sigma \in {K}_{n} \) determines a unique simplicial set map \( {\sigma }_{ * } : \Delta \left\lbrack n\right\rbrack \rightarrow K \) such that \( {\sigma }_{ * }\left( {c}_{n}\right) = \sigma \) . | proof: The verification using (10.2) is straightforward, but the reader may also refer to \( \left\lbrack {122}\right\rbrack \) . | No |
Proposition 10.4 Let \( A \) be a simplicial cochain algebra.\n\n(i) For \( n \geq 0 \) an isomorphism \( A\left( {\Delta \left\lbrack n\right\rbrack }\right) \xrightarrow[]{ \cong }{A}_{n} \) of cochain algebras is given by \( \Phi \mapsto {\Phi }_{{c}_{n}} \), where \( {c}_{n} \) is the fundamental simplex of \( \Del... | proof: (i) The definitions show that \( \Phi \mapsto {\Phi }_{{c}_{n}} \) is a morphism of cochain algebras. Since \( {A}^{p}\left( {\Delta \left\lbrack n\right\rbrack }\right) \) consists of the simplicial set maps \( \Delta \left\lbrack n\right\rbrack \rightarrow {A}^{p} \), Lemma 10.3 asserts that this morphism is a... | Yes |
Proposition 10.5 Suppose \( \theta : D \rightarrow E \) is a morphism of simplicial cochain complexes. Assume that\n\n(i) \( H\left( {\theta }_{n}\right) : H\left( {D}_{n}\right) \rightarrow H\left( {E}_{n}\right) \) is an isomorphism, \( n \geq 0 \).\n\n(ii) \( D \) and \( E \) are extendable.\n\nThen for all simplici... | proof of 10.5: For any inclusion \( L \subset M \) of simplicial sets we have the row exact (Proposition 10.4 (ii)) commutative diagram \n\nHence, if any two of the vertical arrows is a quasi-isomorphism, so is the t... | Yes |
Lemma 10.6 If \( K \) is a simplicial set and \( A \) is an extendable simplicial cochain complex then \( \alpha \) is an isomorphism, natural in \( A \) and in \( K \) . | proof: If \( \alpha \left( \Phi \right) = 0 \) then \( \Phi \) vanishes on all the non-degenerate simplices of \( K\left( n\right) \) and hence \( \Phi = 0 \) . Conversely, given a family \( {\left\{ {\Psi }_{\sigma } \in A\left( \Delta \left\lbrack n\right\rbrack ,\Delta \left\lbrack n - 1\right\rbrack \right) \right\... | Yes |
(i) \( {\left( {A}_{PL}\right) }_{0} = \mathbb{k} \cdot 1 \) . | proof: (i) This is immediate from the definition. | No |
When \( X = \{ {pt}\} \), then \( {S}_{ * }\left( X\right) = \Delta \left\lbrack 0\right\rbrack \) . It follows that \( {A}_{PL}\left( X\right) = \) \( {\left( {A}_{PL}\right) }_{0} = \mathbb{R} \) | \[ {A}_{PL}\left( {pt}\right) = \mathbb{k}. \]\n\nThus an inclusion \( j : {pt} \rightarrow Y \) induces an augmentation \( \varepsilon = {A}_{PL}\left( j\right) : {A}_{PL}\left( Y\right) \rightarrow \) 1k. | No |
Theorem 10.9 [155] Let \( K \) be a simplicial set. Then\n\n(i) There is a natural isomorphism \( {C}_{PL}\left( K\right) \overset{ \cong }{ \rightarrow }{C}^{ * }\left( K\right) \) of cochain algebras.\n\n(ii) The natural morphisms of cochain algebras,\n\n\[ {C}_{PL}\left( K\right) \rightarrow \left( {{C}_{PL} \otimes... | For the proof of Theorem 10.9 we require two lemmas.\n\nLemma 10.11 There are natural isomorphisms \( {C}_{PL}\left( K\right) \overset{ \cong }{ \rightarrow }{C}^{ * }\left( K\right) \).\n\nproof: Each \( \gamma \in {C}_{PL}^{p}\left( K\right), p \geq 0 \) determines the element \( f \in {C}^{p}\left( K\right) \) given... | No |
Corollary 10.10 For topological spaces \( X \) there are natural quasi-isomorphisms of cochain algebras. | \[ {C}^{ * }\left( X\right) \overset{ \simeq }{ \rightarrow }\left( {{C}_{PL} \otimes {A}_{PL}}\right) \left( X\right) \overset{ \simeq }{ \leftarrow }{A}_{PL}\left( X\right) \] This gives the isomorphisms \( {H}^{ * }\left( X\right) \cong H\left( {{A}_{PL}\left( X\right) }\right) \) promised in (10.1). | Yes |
Lemma 10.11 There are natural isomorphisms \( {C}_{PL}\left( K\right) \overset{ \cong }{ \rightarrow }{C}^{ * }\left( K\right) \) . | proof: Each \( \gamma \in {C}_{PL}^{p}\left( K\right), p \geq 0 \) determines the element \( f \in {C}^{p}\left( K\right) \) given by\n\n\[ f\left( \sigma \right) = {\gamma }_{\sigma }\left( {c}_{p}\right) \;,\sigma \in {K}_{p} \]\n\nwhere \( {c}_{p} \) is the fundamental simplex of \( \Delta \left\lbrack p\right\rbrac... | Yes |
(i) \( \oint = \left\{ {\oint }_{n}\right\} : {A}_{PL} \rightarrow {C}_{PL} \) is a quasi-isomorphism of simplicial cochain complexes. | proof: A quasi-isomorphism of extendable simplicial cochain complexes induces a quasi-isomorphism of cochain complexes when applied to any simplicial set (Proposition 10.5). Since \( {C}^{ * }\left( -\right) = {C}_{PL}\left( -\right) \) and since \( {A}_{PL} \) and \( {C}_{PL} \) are extendable (Lemmas 10.11, 10.7 and ... | Yes |
Theorem 11.4 The morphisms \( {\alpha }_{M},{\beta }_{M} \) and \( {\gamma }_{M} \) are all quasi-isomorphisms. In particular, \( {A}_{DR}\left( M\right) \) is weakly equivalent to \( {A}_{PL}\left( {M;\mathbb{R}}\right) \) . | proof of Theorem 11.4: (i) \( {\beta }_{M} \) is a quasi-isomorphism. Lemmas 10 and 11.4 assert that \( {A}_{PL}\left( {-;\mathbb{R}}\right) \rightarrow {A}_{DR} \) is a quasi-isomorphism of extendable simplicial cochain algebras. Thus Proposition 10.5 asserts that \( {\beta }_{M} \) is a quasi-isomorphism.\n\n(ii) \( ... | Yes |
Lemma 11.5 With the hypotheses above, \( {\theta }_{M} \) is a quasi-isomorphism for all smooth \( n \) -manifolds \( M \) . | proof: An \( i \) -basis for \( M \) is a family of open sets \( {V}_{\lambda } \subset M \), closed under finite intersection, and such that any open subset of \( M \) is the union of some of the \( {V}_{\lambda } \) . Given such an \( i \) -basis it is possible to write \( M = O \cup W \) where \( O = \mathop{\coprod... | Yes |
The spheres, \( {S}^{k} \) . Recall that in \( §4\\left( \\mathrm{c}\\right) \) we defined the fundamental class \( \\left\\lbrack {S}^{k}\\right\\rbrack \\in {H}_{k}\\left( {{S}^{k};\\mathbb{Z}}\\right) \). This determines a unique class \( \\omega \\in {H}^{k}\\left( {{A}_{PL}\\left( {S}^{k}\\right) }\\right) \) such... | Suppose, on the other hand, that \( k \) is even. We may still define \( m : \\left( {\\Lambda \\left( e\\right) ,0}\\right) \\rightarrow \) \( {A}_{PL}\\left( {S}^{k}\\right) \) by: \( \\deg e = k,{me} = \\Phi \). But now, because \( \\deg e \) is even, \( \\Lambda \\left( e\\right) \) has as basis \( 1, e,{e}^{2},{e}... | Yes |
Suppose \( {m}_{X} : \left( {{\Lambda V}, d}\right) \rightarrow {A}_{PL}\left( X\right) \) and \( {m}_{Y} : \left( {{\Lambda W}, d}\right) \rightarrow {A}_{PL}\left( Y\right) \) are Sullivan models for path connected topological spaces \( X \) and \( Y \) . Assume further that the rational homology of one of these spac... | In fact, \( {A}_{PL}\left( {p}^{X}\right) \cdot {A}_{PL}\left( {p}^{Y}\right) \) is clearly a morphism of graded vector spaces commuting with the differentials. It is a morphism of algebras because \( {A}_{PL}(X \times \) \( Y \) ) is commutative. To see that it is a quasi-isomorphism use Corollary 10.10 to identify th... | Yes |
Example 4 A cochain algebra \( \left( {{\Lambda V}, d}\right) \) that is not a Sullivan algebra. | Consider the cochain algebra \( \left( {A, d}\right) = \left( {\Lambda \left( {{v}_{1},{v}_{2},{v}_{3}}\right), d}\right) \) , \( \deg {v}_{i} = 1 \), with \( d{v}_{1} = \) \( {v}_{2}{v}_{3}, d{v}_{2} = {v}_{3}{v}_{1} \), and \( d{v}_{3} = {v}_{1}{v}_{2} \) . Here \( \left( {A, d}\right) \) is not a Sullivan algebra. (... | Yes |
Proposition 12.2 Suppose \( \left( {A, d}\right) \) is a commutative cochain algebra such that \( {H}^{0}\left( A\right) = \mathbb{k} \) and \( {H}^{1}\left( A\right) = 0 \) . Then\n\n(i) The morphism \( m : \left( {{\Lambda V}, d}\right) \rightarrow \left( {A, d}\right) \) constructed above is a minimal Sullivan model... | proof: (i) Since \( d : {V}^{k + 1} \rightarrow \Lambda {V}^{ \leq k} \), this exhibits \( \left( {{\Lambda V}, d}\right) \) as a Sullivan algebra. More, \( {\left( \Lambda {V}^{ \leq k}\right) }^{k + 2} \subset {\Lambda }^{ + }{V}^{ \leq k} \cdot {\Lambda }^{ + }{V}^{ \leq k} \), and so \( \left( {{\Lambda V}, d}\righ... | Yes |
Example 6 Simply connected topological spaces \( X \) with finite dimensional homology admit finite dimensional commutative models. | Suppose \( X \) is a simply connected topological space such that \( {H}_{ * }\left( {X;\mathbb{Q}}\right) \) is finite dimensional (e.g. a simply connected finite CW complex — cf. Theorem 4.18). Then \( X \) has a minimal model\n\n\[ m : \left( {{\Lambda V}, d}\right) \overset{ \simeq }{ \rightarrow }{A}_{PL}\left( X\... | Yes |
Example 7 The minimal Sullivan algebra \( \left( {\Lambda \left( {a, b, x, y, z}\right), d}\right) \), where\n\n\[ \n{da} = {db} = 0,\;{dx} = {a}^{2},\;{dy} = {ab},\;{dz} = {b}^{2} \]\n\nand \( \deg a = \deg b = 2 \) and \( \deg x = \deg y = \deg z = 3 \) . | Here, the cohomology algebra \( H \) has as basis\n\n\[ \n1,\;\alpha = \left\lbrack a\right\rbrack ,\;\beta = \left\lbrack b\right\rbrack ,\;\gamma = \left\lbrack {{ay} - {bx}}\right\rbrack ,\;\delta = \left\lbrack {{by} - {az}}\right\rbrack ,\;\varepsilon = \left\lbrack {{aby} - {b}^{2}x}\right\rbrack . \]\n\nNote tha... | No |
Lemma 12.4 (Lifting lemma) There is a morphism \( \varphi : \left( {{\Lambda V}, d}\right) \rightarrow \left( {A, d}\right) \) such that \( {\eta \varphi } = \psi \) ( \( \varphi \) is a lift of \( \psi \) through \( \eta \) ). | proof: We may suppose \( V \) is the increasing union of graded subspaces \( V\left( k\right) \) , \( k \geq 0 \) such that \( V\left( k\right) = V\left( {k - 1}\right) \oplus {V}_{k} \) and \( d : {V}_{k} \rightarrow {\Lambda V}\left( {k - 1}\right) \) . Assume \( \varphi \) is constructed in \( V\left( {k - 1}\right)... | Yes |
Lemma 12.5 \( \varepsilon : \left( {E\left( U\right) ,\delta }\right) \rightarrow \mathbb{k} \) is a quasi-isomorphism; i.e.\n\n\[ H\left( {E\left( U\right) ,\delta }\right) = \mathbb{k}. \] | proof: Let \( \left\{ {u}_{\alpha }\right\} \) be a basis for \( U \) . A direct calculation (using char \( \mathbb{k} = 0 \) if \( \deg {u}_{\alpha } \) is even) shows that \( H\left( {\Lambda \left( {{u}_{\alpha }, d{u}_{\alpha }}\right) }\right) = \mathbb{R} \) . But \( E\left( U\right) = \bigotimes \Lambda \left( {... | No |
Proposition 12.6 If \( {f}_{0} \sim {f}_{1} : X \rightarrow Y \) then \( {A}_{PL}\left( {f}_{0}\right) \psi \sim {A}_{PL}\left( {f}_{1}\right) \psi \) : \( \left( {{\Lambda V}, d}\right) \rightarrow {A}_{PL}\left( X\right) \) | proof: Identify \( \Lambda \left( {t,{dt}}\right) \) as a subcochain algebra of \( {A}_{PL}\left( I\right) \), by mapping \( t \mapsto \) \( u \in {A}_{PL}^{0}\left( I\right) \), where \( u \) restricts to 0 at \( \{ 0\} \) and to 1 at \( \{ 1\} \) . Denote by \( {j}_{0},{j}_{1} \) : \( X \rightarrow X \times I \) the ... | Yes |
Null homotopic morphisms into \( \left( {A,0}\right) \) are constant. | Let \( \left( {{\Lambda V}, d}\right) \) be a minimal Sullivan algebra and let \( \left( {A,0}\right) \) be any commutative cochain algebra with zero differential. The constant morphism \( \varepsilon : \left( {{\Lambda V}, d}\right) \rightarrow \) \( \left( {A,0}\right) \) is defined by \( \varepsilon \left( V\right) ... | Yes |
Since \( {A}_{PL}\left( {pt}\right) = \mathbb{R}\left( {§{10}\left( \mathrm{\;d}\right) }\right) \) the inclusion of a point \( j : x \rightarrow X \) induces an augmentation\n\n\[ \varepsilon = {A}_{PL}\left( j\right) : {A}_{PL}\left( X\right) \rightarrow \mathbb{k}. \]\n\nLet \( \left( {{X}_{\alpha },{x}_{\alpha }}\r... | Thus the first morphism is a quasi-isomorphism: the fibre product of augmented commutative models for the \( {X}_{\alpha } \) is an augmented commutative model for \( {\bigvee }_{\alpha }{X}_{\alpha } \) . | Yes |
Define an automorphism, \( \varphi \), of this model by:\n\n\[ \n{\varphi x} = x,\;{\varphi y} = y,\;{\varphi z} = z,\;{\varphi a} = a,\;{\varphi u} = u + {xyz}. \n\]\n\nThen \( H\left( \varphi \right) = {id} \) and \( Q\left( \varphi \right) = {id} \) but \( \varphi \) is not homotopic to the identity. | Indeed, suppose \( {\varphi }_{0} \sim {\varphi }_{1} : \left( {{\Lambda V}, d}\right) \rightarrow \left( {{\Lambda V}, d}\right) \) via a homotopy, \( \Phi \) . For degree reasons, \( {\Phi a} = a \otimes {f}_{1}\left( t\right) + {xy} \otimes {f}_{2}\left( t\right) + z \otimes {f}_{3}\left( t\right) {dt} \) . Similarl... | Yes |
Example 5 [66] Two morphisms that homotopy commute, but are not homotopic to commuting morphisms. | Define a minimal Sullivan algebra \( \left( {{\Lambda V}, d}\right) \) by specifying the differential in a basis of \( V \) as follows:\n\n\[ \n{V}^{3}\;{V}^{5}\;{V}^{6}\;{V}^{7}\;{V}^{11} \n\]\n\n\[ \n\begin{array}{l} {x}_{1},{x}_{2},{x}_{3}\;y\;z\;v\;{w}_{1},{w}_{2} \\ d{x}_{i} = 0\;{dy} = {x}_{2}{x}_{3}\;{dz} = 0\;{... | Yes |
Now fix \( k \geq 1 \) and any graded vector space \( Z \) of the form \( Z = {\left\{ {Z}^{i}\right\} }_{i \geq 2} \) . We shall construct a minimal Sullivan algebra \( \left( {{\Lambda V}, d}\right) \) homogeneous with respect to word length of degree \( k \) and with the following properties:\n\n- \( V \) is a grade... | For this, set \( {V}_{0} = Z \), and \( d = 0 \) in \( Z \) . Next, construct \( d \) and \( {V}_{m} \) inductively so that \( d : {V}_{m + 1}\overset{ \cong }{ \rightarrow }{\left( {\Lambda }^{k + 1}{V}_{ \leq m}\right) }_{m} \cap \ker d \) . Then \( H\left( d\right) : {V}_{m + 1}\overset{ \cong }{ \rightarrow } \) \(... | No |
Example 7 Sullivan models for cochain algebras \( \left( {H,0}\right) \) with trivial multiplication. | Let \( H = \mathbb{R} \oplus {H}^{ \geq 2} \) be a graded algebra with trivial multiplication: \( {H}^{ + } \cdot {H}^{ + } = \) 0 . Regard \( H \) as a cochain algebra with zero differential.\n\nIn Example 6 we constructed word-length homogeneous Sullivan algebras. Here we consider the case \( k = 1 \) and \( Z = {H}^... | Yes |
The cochain algebra \( {A}_{DR}{\left( G\right) }^{G} \) of right invariant forms on \( G \), and its minimal model. | Let \( \mathfrak{g} \) be the Lie algebra of the group \( G \), identified with the Lie algebra of right invariant vector fields on \( G \) . Regard \( {\mathfrak{g}}^{\sharp } = \operatorname{Hom}\left( {\mathfrak{g};\mathbb{R}}\right) \) as a graded vector space concentrated in degree 1 . Let \( G \) act on itself by... | Yes |
Let \( G \) be a nilpotent connected Lie group and let \( \Gamma \) be a discrete sub-group of \( G \) such that the quotient space \( X = G/\Gamma \) is compact. The covering projection \( \pi : G \rightarrow X \) induces an isomorphism \( {A}_{DR}X \cong {\left( {A}_{DR}G\right) }^{\Gamma }( = \) the complex of right... | In this case the Lie algebra \( \mathfrak{g} \) of \( G \) is nilpotent and so \( \left( {\Lambda {\mathfrak{g}}^{\sharp }, d}\right) \) is a minimal model of \( X \) over the real numbers. Obviously, \( d = 0 \) if and only if the Lie algebra \( \mathfrak{g} \) is abelian. In this case \( X = {S}^{1} \times \cdots \ti... | No |
Example 3 Symmetric spaces are formal. | Suppose \( \tau \) is an involution of a compact connected Lie group \( G \), and that \( K \) is the connected component of the identity in the subgroup of elements fixed by \( \tau \) . Then \( G/K \) is called a symmetric space of compact type. By Cartan’s theorem, \( {A}_{DR}{\left( G/K\right) }^{G}\overset{ \simeq... | Yes |
Example 4 (Deligne, Griffiths, Morgan and Sullivan [42]) Compact Kähler manifolds are formal. | Given a Kähler manifold, put \( {d}^{c} = {J}^{-1}{dJ} : {A}_{DR}\left( M\right) \rightarrow {A}_{DR}\left( M\right) \) . Then \( {\left( {d}^{c}\right) }^{2} = 0 \) and \( {d}^{c}d = d{d}^{c} \) . In [42] the authors show that \( d : \ker {d}^{c} \rightarrow \operatorname{Im}{d}^{c} \) and that the obvious inclusion a... | Yes |
Lemma 13.3 If \( \gamma ,\beta ,\alpha \) are quasi-isomorphisms and if one of \( \varphi ,\psi \) and one of \( {\varphi }^{\prime },{\psi }^{\prime } \) are surjective then \( \left( {\gamma ,\alpha }\right) : \left( {C{ \times }_{B}A, d}\right) \rightarrow \left( {{C}^{\prime }{ \times }_{{B}^{\prime }}{A}^{\prime }... | proof: If \( \psi \) and \( {\psi }^{\prime } \) are both surjective then \( \alpha \) restricts to a quasi-isomorphism \( \ker \psi \overset{ \simeq }{ \rightarrow }\ker {\psi }^{\prime } \) . Now use the row exact diagram\n\n are quasi-isomorphisms and that one of \( \varphi \) and \( \psi \) and one of \( \xi \) and \( \eta \) are surjective. Then the fibre squares corresponding to \( \varphi ,\psi \) and to \( \xi ,\eta \) are weakly equivalent. | proof: For definiteness take \( \psi \) to be surjective. Suppose first that both \( \xi \) and \( \eta \) are surjective. Then we construct morphisms \( {\alpha }^{\prime } \sim \alpha \) and \( {\gamma }^{\prime } \sim \gamma \) such that replacing \( \alpha \) by \( {\alpha }^{\prime } \) and \( \gamma \) by \( {\ga... | Yes |
Proposition 13.5 If \( {H}_{ * }\left( {Z, Y;\mathbb{k}}\right) \overset{ \cong }{ \rightarrow }{H}_{ * }\left( {X{ \cup }_{f}Z, X;\mathbb{k}}\right) \) then the morphism \( \left( {{A}_{PL}\left( {i}_{X}\right) ,{A}_{PL}\left( {f}_{Z}\right) }\right) \) is a quasi-isomorphism. Thus the fibre product is a commutative m... | proof: Since \( {H}_{ * }\left( {Z, Y;\mathbb{R}}\right) \overset{ \cong }{ \rightarrow }{H}_{ * }\left( {X{ \cup }_{f}Z, X;\mathbb{R}}\right) \) it follows that \( {A}_{PL}\left( {X{ \cup }_{f}}\right. \) \( Z, X)\overset{ \simeq }{ \rightarrow }{A}_{PL}\left( {Z, Y}\right) \) . Thus in the row exact diagram\n\n![4c8f... | Yes |
Proposition 13.6 If \( {H}_{ * }\left( {Z, Y;\mathbb{k}}\right) \overset{ \cong }{ \rightarrow }{H}_{ * }\left( {X{ \cup }_{f}Z, X;\mathbb{k}}\right) \) and if one of \( \varphi ,\psi \) is surjective then \( \mathcal{D} \) is weakly equivalent to the fibre product square | proof: This is an immediate translation of Lemma 13.4, given Proposition 13.5. | No |
Proposition 13.8 The fibre product square for (13.7) is weakly equivalent to the commutative adjunction space for \( X{ \cup }_{f}{CY} \) . In particular the cochain algebra \( {\Lambda V}{ \times }_{B}\left( {\mathbb{R} \oplus \left\lbrack {B \otimes {\Lambda }^{ + }\left( {t,{dt}}\right) }\right\rbrack }\right) \) is... | proof: Use Proposition 13.5 to identify \( {A}_{PL}\left( X\right) { \times }_{{A}_{PL}\left( Y\right) }{A}_{PL}\left( {CY}\right) \) as a commutative model for \( X{ \cup }_{f}{CY} \) . Let \( {i}_{1} \) denote the inclusion \( y \mapsto \left( {y,1}\right) \) of \( Y \) in \( {CY} \) and in \( Y \times I \) and also ... | Yes |
Proposition 13.9 The suspension, \( {\sum Y} \), of a well-based topological space \( \left( {Y,{y}_{0}}\right) \) is formal, and satisfies \( {H}^{ + }\left( {{\sum Y};\mathbf{k}}\right) \cdot {H}^{ + }\left( {{\sum Y};\mathbf{k}}\right) = 0 \) . | proof: Apply Proposition 13.8 to the case of the constant map \( f : Y \rightarrow \{ {pt}\} \) to obtain a commutative model for \( \{ {pt}\} { \cup }_{f}{CY} \) of the form \( \mathbb{K}{ \times }_{B}\left( {\mathbb{R} \oplus \left\lbrack {B \otimes {\Lambda }^{ + }\left( {t,{dt}}\right) }\right\rbrack }\right) \) . ... | Yes |
Proposition 13.12 The cochain algebra \( \left( {{\Lambda V} \oplus \mathbb{R}u,{d}_{\alpha }}\right) \) is a commutative model for \( X{ \cup }_{a}{D}^{n + 1} \) . | proof: We apply Proposition 13.8, noting that \( {D}^{n + 1} = C{S}^{n} \) . This gives a commutative model for \( X{ \cup }_{a}{D}^{n + 1} \) of the form \( {\Lambda V}{ \times }_{\Lambda W}\left( {\mathbb{k} \oplus \left\lbrack {{\Lambda }^{ + }W \otimes {\Lambda }^{ + }\left( {t,{dt}}\right) }\right\rbrack }\right) ... | Yes |
The even spheres \( {S}^{2n} \). | Let \( \alpha \in {\pi }_{2n}\left( {S}^{2n}\right) \) be represented by the identity map, and recall that the minimal Sullivan model of \( {S}^{2n} \) has the form \( \left( {\Lambda \left( {e,{e}^{\prime }}\right), d{e}^{\prime } = {e}^{2}}\right) \). It is clear that \( \langle e;\alpha \rangle = 1 \), and so\n\n\[ ... | Yes |
Example 2 \( \;{S}^{3} \vee {S}^{3}{ \cup }_{f}\left( {{D}_{0}^{8} \coprod {D}_{1}^{8}}\right) \) . | Let \( {a}_{0},{a}_{1} : {S}^{3} \rightarrow {S}^{3} \vee {S}^{3} \) denote the inclusions of the left and right hand spheres, and put \( {\alpha }_{i} = \left\lbrack {a}_{i}\right\rbrack \in {\pi }_{3}\left( {{S}^{3} \vee {S}^{3}}\right) \) . Attach \( {D}_{0}^{8} \) and \( {D}_{1}^{8} \) to \( {S}^{3} \vee {S}^{3} \)... | Yes |
Lemma 14.1 \( \left( {B \otimes {\Lambda V}, d}\right) \) is \( \left( {B, d}\right) \) -semifree. | proof: Write \( V = \mathop{\bigcup }\limits_{{k \geq 0}}V\left( k\right) \) as in the definition, and set \( V\left( {-1}\right) = 0 \) . Write \( V\left( k\right) = V\left( {k - 1}\right) \oplus {V}_{k} \), and simplify notation by writing \( B \otimes {\Lambda V}\left( k\right) = B\left( k\right) \) . Then \( B\left... | Yes |
Proposition 14.3 A morphism \( \varphi : \left( {B, d}\right) \rightarrow \left( {C, d}\right) \) of commutative cochain algebras has a Sullivan model if \( {H}^{0}\left( B\right) = \mathbb{k} = {H}^{0}\left( C\right) \) and \( {H}^{1}\left( \varphi \right) \) is injective. | proof: Choose a graded subspace \( {B}_{1} \subset B \) so that\n\n\[ \n{\left( {B}_{1}\right) }^{0} = \mathbb{k},{\left( {B}_{1}\right) }^{1} \oplus d\left( {B}^{0}\right) = {B}^{1}\;\text{ and }\;{\left( {B}_{1}\right) }^{n} = {B}^{n}, n \geq 2. \n\] \n\nClearly \( \left( {{B}_{1}, d}\right) \) is a sub cochain algeb... | Yes |
Lemma 14.5 If \( {\varphi }_{0} \sim {\varphi }_{1} \) rel \( B \) then \( {\varphi }_{0} - {\varphi }_{1} = {hd} + {dh} \), where \( h : B \otimes {\Lambda V} \rightarrow A \) is a B-linear map of degree -1 . In particular, \( H\left( {\varphi }_{0}\right) = H\left( {\varphi }_{1}\right) \) . | proof: Let \( \Phi : \left( {B \otimes {\Lambda V}, d}\right) \rightarrow \left( {A, d}\right) \otimes \Lambda \left( {t,{dt}}\right) \) be a homotopy rel \( B \) from \( {\varphi }_{0} \) to \( {\varphi }_{1} \) . As in the proof of Proposition 12.8, define \( h : B \otimes {\Lambda V} \rightarrow A \) by\n\n\[ \Phi \... | Yes |
Lemma 14.7 The morphism \( \sigma : B \otimes {\Lambda W} \rightarrow B \otimes {\Lambda V} \) is an isomorphism. | proof: It follows from the hypothesis that \( {V}^{n} \subset {W}^{n} + B \otimes \Lambda {V}^{ < n}, n \geq 1 \), and an obvious induction then gives that \( \sigma \) is surjective. Choose \( \alpha : V \rightarrow B \otimes {\Lambda W} \) so that \( {\sigma \alpha } = {id} \), and extend \( i{d}_{B} \) and \( \alpha... | Yes |
Lemma 14.8 There is a relative Sullivan algebra \( \left( {{B}_{1} \otimes {\Lambda V},{d}^{\prime }}\right) \) and an isomorphism, \[ \sigma : \left( {B \otimes {\Lambda V},{d}^{\prime }}\right) = \left( {B, d}\right) { \otimes }_{\left( {B}_{1}, d\right) }\left( {{B}_{1} \otimes {\Lambda V},{d}^{\prime }}\right) \ove... | proof: Write \( V = \mathop{\bigcup }\limits_{{k \geq 0}}V\left( k\right) \) with \[ \left( {B \otimes {\Lambda V}\left( k\right), d}\right) = \left( {B \otimes {\Lambda V}\left( {k - 1}\right) \otimes \Lambda {V}_{k}, d}\right) \] and \( d : {V}_{k} \rightarrow B \otimes {\Lambda V}\left( {k - 1}\right) \) . Assume by... | Yes |
Theorem 14.11 If \( \eta \) restricts to an isomorphism \( {\eta }_{B} : {B}^{\prime }\overset{ \cong }{ \rightarrow }B \) then \( \eta \) itself is an isomorphism. | proof: Consider the diagram\n\n\n\nWe apply an argument of Gomez-Tato [65] to extend \( {\eta }_{B}^{-1} \) to a morphism \( \gamma \) : \( \left( {B \otimes {\Lambda V}, d}\right) \rightarrow \left( {{B}^{\prime } \... | Yes |
Theorem 14.12 The morphism \( \varphi \) has a minimal Sullivan model\n\n\[ m : \left( {B \otimes {\Lambda V}, d}\right) \overset{ \simeq }{ \rightarrow }\left( {C, d}\right) . \]\n\nIf \( {m}^{\prime } : \left( {B \otimes \Lambda {V}^{\prime }, d}\right) \overset{ \simeq }{ \rightarrow }\left( {C, d}\right) \) is a se... | proof: In Proposition 14.3 we showed \( \varphi \) had a Sullivan model, and in Theorem 14.9 we showed that this is the tensor product of a contractible algebra and a minimal relative Sullivan algebra. Thus \( \varphi \) has a minimal Sullivan model.\n\nGiven two such models we may apply Proposition 14.6 to the diagram... | Yes |
Proposition 14.13 If \( \varphi : \left( {{\Lambda V}, d}\right) \rightarrow \left( {{\Lambda W}, d}\right) \) is a morphism of Sullivan algebras then \( \varphi \) is a quasi-isomorphism if and only if \( H\left( {Q\left( \varphi \right) }\right) \) is an isomorphism. | proof: Write \( \left( {{\Lambda V}, d}\right) = \left( {\Lambda \bar{V}, d}\right) \otimes \left( {E, d}\right) \) and \( \left( {{\Lambda W}, d}\right) = \left( {\Lambda \bar{W}, d}\right) \otimes \left( {F, d}\right) \) with \( \left( {\Lambda \bar{V}, d}\right) \) and \( \left( {\Lambda \bar{W}, d}\right) \) minima... | Yes |
Example 2 A commutative model for a Sullivan fibre. | Let \( \varphi : \left( {B, d}\right) \rightarrow \left( {A, d}\right) \) be a morphism of commutative cochain algebras both of which satisfy \( {H}^{0}\left( -\right) = \mathbb{R} \) and \( {H}^{1}\left( -\right) = 0 \) . Extend \( \varphi \) to a minimal Sullivan model \( m : \left( {B \otimes {\Lambda W}, d}\right) ... | Yes |
Theorem 15.3 Suppose \( Y \) is simply connected and one of the graded spaces \( {H}_{ * }\left( {Y;\mathbb{R}}\right) ,{H}_{ * }\left( {F;\mathbb{R}}\right) \) has finite type. Then\n\n\[ \bar{m} : \left( {{\Lambda V},\bar{d}}\right) \rightarrow {A}_{PL}\left( F\right) \]\n\nis a quasi-isomorphism. | proof: Suppose first that \( p \) is a fibration. We wish to apply Theorem 7.10, with diagram (15.2) corresponding to diagram (7.9). For this we need to verify two things: first, \( m \) has to be an \( {A}_{PL}\left( Y\right) \) -semifree resolution and second, diagram (15.1) has to be weakly equivalent to the corresp... | Yes |
Proposition 15.5 The three morphisms \( {m}_{Y}, m \) and \( \bar{m} \) in (15.4) are all Sullivan models. | proof: (i) The morphism \( {m}_{Y} \) . This is a Sullivan model by hypothesis.\n\n(ii) The morphism \( m \) . This is a quasi-isomorphism by construction.\n\nThus we have only to exhibit \( \left( {\Lambda {V}_{Y} \otimes {\Lambda V}, d}\right) \) as a Sullivan algebra. Put \( W = \) \( {V}_{Y} \oplus V \) and define ... | No |
Proposition 15.6 If \( \bar{n} \) is a quasi-isomorphism then so is \( n \), i.e., | proof: In the proof of Proposition 15.5 we showed that \( \left( {\Lambda {V}_{Y} \otimes {\Lambda W}, d}\right) \) was a Sullivan algebra. Thus it suffices to show that \( n \) is a quasi-isomorphism.\n\nLet \( m : \left( {\Lambda {V}_{Y} \otimes {\Lambda V}, d}\right) \overset{ \simeq }{ \rightarrow }{A}_{PL}\left( X... | Yes |
The model of the loop space \( \Omega {S}^{k}, k \geq 2 \) . | Let \( p : P{S}^{k} \rightarrow {S}^{k} \) be the path space fibration, with fibre \( \Omega {S}^{k} \) . If \( k \) is odd the minimal Sullivan model for \( {S}^{k} \) has the form \( {m}_{S} : \left( {\Lambda \left( e\right) ,0}\right) \overset{ \simeq }{ \rightarrow }{A}_{PL}\left( {S}^{k}\right) \) . Define\n\n\[ m... | Yes |
The rational homotopy type of \( K\left( {\mathbb{Z}, n}\right) \) . | Let \( K\left( {\mathbb{Z}, n}\right) \) denote an Eilenberg-MacLane space of type \( \left( {\mathbb{Z}, n}\right) \), and let \( {a}_{n} \) : \( {S}^{n} \rightarrow K\left( {\mathbb{Z}, n}\right) \) represent a generator of \( {\pi }_{n}\left( {K\left( {\mathbb{Z}, n}\right) }\right) = \mathbb{Z} \) . Then by the Hur... | Yes |
A spherical fibration is a fibration \( p : X \rightarrow Y \) whose fibre has the homotopy type of a sphere \( {S}^{k} \) . Suppose given such a fibration with simply connected base \( Y \) . | If \( k \) is odd then the minimal model of \( {S}^{k} \) has the form \( \left( {\Lambda \left( e\right) ,0}\right) \) . Hence we can apply Theorem 15.3 to obtain a model for \( p \) of the form\n\n\[ \left( {{A}_{PL}\left( Y\right) \otimes \Lambda \left( e\right), d}\right) \overset{ \simeq }{ \rightarrow }{A}_{PL}\l... | Yes |
From the long exact homotopy sequence deduce that \( \mathbb{C}{P}^{\infty } \simeq K\left( {\mathbb{Z},2}\right) \) and hence that \( {H}^{ * }\left( {\mathbb{C}{P}^{\infty };\mathbb{Q}}\right) = {\Lambda u} \), with \( \deg u = 2 \) (Example 2, above). | Similarly, \( {\pi }_{ * }\left( {\mathbb{C}{P}^{n}}\right) \otimes \mathbb{Q} = \mathbb{Q}u \oplus \mathbb{Q}x \) with \( \deg u = 2 \) and \( \deg x = {2n} + 1 \) . Since \( \mathbb{C}{P}^{n} \) is a \( {2n} \) -dimensional CW complex it has no cohomology in degree \( {2n} + 2 \) . Thus its minimal Sullivan model mus... | No |
The free loop space \( {X}^{{S}^{1}} \) . Let \( X \) be a simply connected topological space with rational homology of finite type. The free loop space, \( {X}^{{S}^{1}} \), is the topological space of all continuous maps \( {S}^{1} \rightarrow X \) . We may identify these as the continuous maps \( f : I \rightarrow X... | Moreover \n\n\[ p\left( g\right) = g\left( 0\right) ,\]\n\n\[ q\left( f\right) = \left( {f\left( 0\right), f\left( 1\right) }\right)\]\n\n\[ \Delta \left( x\right) = \left( {x, x}\right) \]\n\nis a pullback diagram o... | No |
Theorem 15.11 Suppose \( X \) is simply connected and \( {H}_{ * }\left( {X;\mathbb{k}}\right) \) has finite type. Then the bilinear map \( {V}_{X} \times {\pi }_{ * }\left( X\right) \rightarrow \mathbb{k} \) is non-degenerate. Equivalently, \[ {\nu }_{X} : {V}_{X}\overset{ \cong }{ \rightarrow }{\operatorname{Hom}}_{\... | ## proof of Theorem 15.11: Fix \( k \geq 2 \) . To show that \( {\nu }_{X} : {V}_{X}^{k}\overset{ \cong }{ \rightarrow }{\operatorname{Hom}}_{\mathbb{Z}}\left( {{\pi }_{k}\left( X\right) ,\mathbb{R}}\right) \) we let \( r \) be the least integer such that \( {\pi }_{r}\left( X\right) \neq 0 \), and argue by induction o... | No |
Let \( \iota \in {\pi }_{n}\left( {S}^{n}\right) \) be the class represented by the identity map of \( {S}^{n} \) . Then \[ {\pi }_{n}\left( {S}^{{2k} + 1}\right) \otimes \mathbb{Q} = \left\{ \begin{array}{ll} \mathbb{Q} \cdot \iota &, n = {2k} + 1 \\ 0 & ,\text{ otherwise } \end{array}\right. \] and \[ {\pi }_{n}\left... | Indeed in the first case the minimal model is \( \left( {\Lambda \left( e\right) ,0}\right) \) and \( \langle e;\iota \rangle = 1 \) . In the second the minimal model is \( \left( {\Lambda \left( {e,{e}^{\prime }}\right), d{e}^{\prime } = {e}^{2}}\right) \) . Again \( \langle e;\iota \rangle = 1 \) while Proposition 13... | Yes |
Example 2 The model \( \left( {\Lambda \left( {{e}_{0},{e}_{1}, x}\right) ,{dx} = {e}_{0}{e}_{1}}\right), k = \mathbb{Q} \) . | In Example 2 of \( §{13}\left( \mathrm{e}\right) \) we considered the space \( X = \left( {{S}^{3} \vee {S}^{3}}\right) \mathop{\bigcup }\limits_{f}\left( {{D}_{0}^{8}\text{II}}\right. \) \( \left. {D}_{1}^{8}\right) \) where the two 8-cells were attached respectively by \( {\left\lbrack {a}_{0},{\left\lbrack {a}_{0},{... | Yes |
Example 3 The Quillen plus construction . Let \( X \) be a path connected topological space whose fundamental group, \( {\pi }_{1}\left( X\right) \) is finitely generated and such that every element in \( {\pi }_{1}\left( X\right) \) is a product of commutators. Then \( {H}_{1}\left( {X;\mathbb{Z}}\right) = 0 \), by th... | Adjoin to \( X \) finitely many two cells \( {e}_{1}^{2},\ldots ,{e}_{n}^{2} \) to kill a set of generators of \( {\pi }_{1}\left( X\right) \) . The Cellular approximation theorem 1.2 implies that \( {\pi }_{1}\left( X\right) \rightarrow \) \( {\pi }_{1}\left( {X \cup \mathop{\bigcup }\limits_{i}{e}_{i}^{2}}\right) \) ... | Yes |
Let \( X \) be a smooth manifold. In \( §{11} \) we showed that \( {A}_{DR}\left( X\right) \) is connected by natural quasi-isomorphisms to \( {A}_{PL}\left( {X;\mathbb{R}}\right) \) . Thus a minimal Sullivan model \( \left( {{\Lambda V}, d}\right) \overset{ \simeq }{ \rightarrow }{A}_{DR}\left( X\right) \) is a Sulliv... | \[ {\nu }_{X} : V\overset{ \cong }{ \rightarrow }{\operatorname{Hom}}_{\mathbb{Z}}\left( {{\pi }_{ * }\left( X\right) ,\mathbb{R}}\right) \] (Theorem 15.11). This shows that the real homotopy groups, as well as the cohomology algebra, may be computed from \( {A}_{DR}\left( X\right) \) . | Yes |
Proposition 15.14 The linear maps \( {d}_{0}\zeta \) and \( {hu}{r}_{F}{\partial }_{ * } \) are dual, up to sign, if \( F \) is path connected and \( Y \) is simply connected. | proof: Let \( z \in {\left( \Lambda V\right) }^{k} \) be a \( \bar{d} \) -cocycle and let \( \alpha \in {\pi }_{k + 1}\left( Y\right) \) be represented by \( a : {S}^{k + 1} \rightarrow Y \) . We show that\n\n\[ \left\langle {{d}_{0}{\zeta z};\alpha }\right\rangle = {\left( -1\right) }^{k + 1}\left\langle {H\left( \bar... | Yes |
Proposition 15.15 The minimal Sullivan model for \( {B}_{G} \) has the form\n\n\[ \n{m}_{{B}_{G}} : \left( {\Lambda {V}_{{B}_{G}},0}\right) \overset{ \simeq }{ \rightarrow }{A}_{PL}\left( {B}_{G}\right)\n\]\n\nwhere \( {V}_{{B}_{G}}^{ * } \cong {P}_{G}^{* + 1} \) . In particular, \( {H}^{ * }\left( {B}_{G}\right) \) is... | Since \( G \) is weakly equivalent to \( \Omega {B}_{G},{B}_{G} \) is simply connected. Since \( \left( {\Lambda {V}_{{B}_{G}},0}\right) \) is the Sullivan model of \( {B}_{G} \), it follows from Theorem 15.11 that \( {V}_{{B}_{G}} \cong \) \( {\operatorname{Hom}}_{\mathbb{Z}}\left( {{\pi }_{ * }\left( {B}_{G}\right) ,... | Yes |
Let \( K \subset G \) be a closed subgroup of a connected Lie group \( G \) . Right multiplication by \( K \) is an action on \( G \) and the projection \( p : G \rightarrow G/K \) onto the orbit space is the projection of a principal \( K \) -bundle [70]. The space (in fact, a smooth manifold) \( G/K \) is called a ho... | Proposition 15.16 The Sullivan algebra \( \left( {\Lambda {V}_{{B}_{K}} \otimes \Lambda {P}_{G}, d}\right) \) defined by \( d{x}_{i} = \) \( {H}^{ * }\left( {B\left( j\right) }\right) {y}_{i} \) and \( d = 0 \) in \( {V}_{{B}_{K}} \) is a Sullivan model for \( G/K \) . | No |
Proposition 15.17 If a compact connected Lie group acts smoothly and freely on a manifold \( X \) then the Sullivan algebra \( \left( {\Lambda {V}_{{B}_{G}} \otimes \Lambda {V}_{X}, D}\right) \) is a Sullivan model for the orbit space \( X/G \) . In particular, \( {H}^{i}\left( {\Lambda {V}_{{B}_{G}} \otimes \Lambda {V... | proof: In this case the projection \( X \rightarrow X/G \) is the projection of a principal \( G \) bundle [REF] and Proposition 2.9 provides a weak homotopy equivalence \( {q}^{\prime } : {X}_{G} \rightarrow X/G \) . This implies the first assertion. Since \( X/G \) is a manifold and \( \dim X/G = \dim X - \dim G \), ... | Yes |
Proposition 15.17 If a compact connected Lie group acts smoothly and freely on a manifold \( X \) then the Sullivan algebra \( \left( {\Lambda {V}_{{B}_{G}} \otimes \Lambda {V}_{X}, D}\right) \) is a Sullivan model for the orbit space \( X/G \) . In particular, \( {H}^{i}\left( {\Lambda {V}_{{B}_{G}} \otimes \Lambda {V... | proof: In this case the projection \( X \rightarrow X/G \) is the projection of a principal \( G \) bundle [REF] and Proposition 2.9 provides a weak homotopy equivalence \( {q}^{\prime } : {X}_{G} \rightarrow X/G \) . This implies the first assertion. Since \( X/G \) is a manifold and \( \dim X/G = \dim X - \dim G \), ... | Yes |
Example 4 A fibre bundle with fibre \( {S}^{3} \times {SU}\left( 3\right) \) that is not principal. | Consider the continuous map\n\n\[ f : {S}^{3} \times {S}^{3} \rightarrow \left( {{S}^{3} \times {S}^{3}}\right) /\left( {{S}^{3} \vee {S}^{3}}\right) = {S}^{6}\overset{b}{ \rightarrow }{B}_{{SU}\left( 3\right) } \]\n\nand use it to pull the universal bundle back to a principal \( {SU}\left( 3\right) \) -bundle \( p \) ... | Yes |
Consider the manifold \( X \) of Example 4. We observed there that a certain bundle \( q : X \rightarrow {S}^{3} \) with fibre \( {S}^{3} \times {SU}\left( 3\right) \) was not principal. Now we establish a stronger assertion: \( X \) does not admit any free smooth \( {S}^{3} \times {SU}\left( 3\right) \) action. | In fact we show more: \( X \) has no free smooth \( {S}^{3} \times {S}^{3} \) action. Indeed let \( G = {S}^{3} \times {S}^{3} \) . For any \( G \) action the Borel construction has a Sullivan model of the form\n\n\[ \left( {\Lambda \left( {{a}_{1},{a}_{2}}\right) \otimes \Lambda \left( {u, v,{x}_{2},{x}_{3}}\right), D... | Yes |
Lemma 16.3 If \( Y = {\Omega X} \) then \( H\left( \Delta \right) : {H}_{ * }\left( {{\Omega X};\mathbf{k}}\right) \rightarrow {H}_{ * }\left( {{\Omega X};\mathbf{k}}\right) \otimes {H}_{ * }\left( {{\Omega X};\mathbf{k}}\right) \) is a morphism of graded algebras. | proof: Observe that \( {\Omega X} \times {\Omega X} \) is a topological monoid with component-wise multiplication, and that \( {\Delta }_{\text{top }} \) is a morphism of topological monoids. It follows that \( {H}_{ * }\left( {\Delta }_{\text{top }}\right) : {H}_{ * }\left( {{\Omega X};\mathbf{k}}\right) \rightarrow {... | Yes |
The homology algebra \( {H}_{ * }\left( {\Omega {S}^{{2k} + 1};\mathbb{k}}\right), k \geq 1 \) . | Let \( f : {S}^{2k} \rightarrow \Omega {S}^{{2k} + 1} \) satisfy \( \left\lbrack f\right\rbrack = {\partial }_{ * }\left\lbrack {i{d}_{{S}^{{2k} + 1}}}\right\rbrack \) . Then the Hurewicz theorem 4.19 asserts that \( {H}_{ * }\left( f\right) \left\lbrack {S}^{2k}\right\rbrack \) is a non-zero homology class \( \alpha \... | Yes |
Proposition 16.6 The quadratic part of the differential in \( \Lambda {V}_{X} \) is given by\n\n\[ \n{d}_{1}{d}_{0}v = \mathop{\sum }\limits_{i}{\left( -1\right) }^{\deg {\Phi }_{i}}\left( {{d}_{0}\zeta {\Phi }_{i}}\right) \land \left( {{d}_{0}\zeta {\Psi }_{i}}\right) .\n\] | proof: Write \( {dv} = {d}_{0}v + \mathop{\sum }\limits_{i}{u}_{i} \otimes {v}_{i} + \Phi + \Omega \) with \( {u}_{i} \in {V}_{X},{v}_{i} \in V,\Phi \in {\Lambda }^{2}{V}_{X} \) , and \( \Omega \in {\Lambda }^{ \geq 3}\left( {{V}_{X} \oplus V}\right) \) . Since \( {d}^{2}v = 0 \) the component of \( {d}^{2}v \) in \( {... | Yes |
Proposition 16.8 The distinguished homology classes \( {\beta }_{{i}_{1}}\cdots \cdots {\beta }_{{i}_{p}} \cdot {\alpha }_{{j}_{1}}^{{n}_{1}}\cdots \cdots {\alpha }_{{j}_{q}}^{{n}_{q}} \) are a basis of the graded vector space \( {H}_{ * }\left( {{\Omega X};\mathbb{k}}\right) \) . | proof: Recall that \( {\bar{a}}_{j} \sim \left( {\Omega {a}_{j}}\right) \overline{id} \), where \( \overline{id} : {S}^{2{k}_{j}} \rightarrow \Omega {S}^{2{k}_{j} + 1} \) represents \( {\partial }_{ * }\left\lbrack {i{d}_{{S}^{2{k}_{j} + 1}}}\right\rbrack \) . Denote \( {H}_{ * }\left( \overline{id}\right) \left\lbrack... | Yes |
Theorem 16.10 (Cartan-Serre) The Hurewicz homomorphism extends to an isomorphism | proof: We begin with three simple observations.\n\n(i) hur: \( {\pi }_{ * }\left( {S}^{2\ell + 1}\right) \otimes \mathbf{k}\overset{ \cong }{ \rightarrow }{P}_{ * }\left( {{S}^{2\ell + 1};\mathbf{k}}\right) ,\ell \geq 0 \) .\n\nIndeed, the Hurewicz theorem 4.19 asserts this is an isomorphism in degrees \( \leq 2\ell + ... | No |
Proposition 16.11 If \( {\gamma }_{0} \in {\pi }_{k + 1}\left( X\right) \) and \( {\gamma }_{1} \in {\pi }_{n + 1}\left( X\right) \) then\n\n\[ \left\lbrack {\theta {\gamma }_{0},\theta {\gamma }_{1}}\right\rbrack = {\left( -1\right) }^{k + 1}\theta \left( {\left\lbrack {\gamma }_{0},{\gamma }_{1}\right\rbrack }_{W}\ri... | proof: In Proposition 16.6 we showed that the quadratic part of the differential in \( \Lambda {V}_{X} \) was given by\n\n\[ {d}_{1}{d}_{0}v = \mathop{\sum }\limits_{i}{\left( -1\right) }^{\deg {\Phi }_{i}}\left( {{d}_{0}\zeta {\Phi }_{i}}\right) \land \left( {{d}_{0}\zeta {\Psi }_{i}}\right) ,\;v \in V, \]\n\nwhere \(... | Yes |
Theorem 16.13 The Hurewicz homomorphism extends uniquely to an isomorphism of graded algebras | \[ \left( {T{L}_{X}}\right) /I\overset{ \cong }{ \rightarrow }{H}_{ * }\left( {{\Omega X};\mathbf{k}}\right) \] proof: First observe that an identification \( {L}_{X} = {\pi }_{ * }\left( {\Omega X}\right) \otimes \mathbb{k} \) is specified by identifying \( x \) with \( {\partial }_{ * }{sx}, x \in {L}_{X} \) . This i... | Yes |
The homology algebra \( {H}_{ * }\left( {\Omega {S}^{n + 1};\mathbf{k}}\right) \) . | Let \( a : {S}^{n} \rightarrow \Omega {S}^{n + 1} \) represent \( {\partial }_{ * }\left\lbrack {i{d}_{{S}^{n + 1}}}\right\rbrack \), and let \( \alpha = \operatorname{hur}\left\lbrack a\right\rbrack = {H}_{ * }\left( a\right) \left\lbrack {S}^{n}\right\rbrack \in \) \( {H}_{n}\left( {\Omega {S}^{n};\mathbb{R}}\right) ... | Yes |
Lemma 17.1 [122] The quotient map \( {q}_{K} : \mathop{\coprod }\limits_{n}{K}_{n} \times {\Delta }^{n} \rightarrow \left| K\right| \) restricts to a bijection\n\n\[{\widetilde{q}}_{K} : \mathop{\coprod }\limits_{n}N{K}_{n} \times {\overset{ \circ }{\Delta }}^{n} \rightarrow \left| K\right| . | proof: Denote \( {q}_{K} \) simply by \( q \) . If \( \sigma \times x \in {K}_{n} \times {\Delta }^{n} \) choose the least \( k \) such that \( q\left( {\sigma \times x}\right) = q\left( {\tau \times y}\right) \) for some \( \left( {\tau, y}\right) \in {K}_{k} \times {\Delta }^{k} \) . Then \( \tau \) cannot be degener... | No |
The Milnor realization \( \left| K\right| \) of a simplicial set \( K \) is a \( {CW} \) complex with \( n \) -skeleton \( \left| {K\left( n\right) }\right| \) and \( n \) -cells the non-degenerate \( n \) -simplices \( \sigma \in N{K}_{n} \) . The attaching map for \( \sigma \) is the restriction of \( {q}_{K} \) to \... | proof: Let \( \sigma \in {K}_{n} \) . The quotient map \( q : \mathop{\coprod }\limits_{n}{K}_{n} \times {\Delta }^{n} \rightarrow \left| K\right| \) satisfies \( q(\sigma \times \) \( \left. {{\lambda }_{i}x}\right) = q\left( {{\partial }_{i}\sigma \times x}\right) \) and it follows that \( q \) restricts to a continu... | Yes |
The ’simplicial point’ is the unique simplicial set with a single \( k \) -simplex \( {c}_{n} \) in each dimension. Now suppose \( K \) is any simplicial set. We extend the face and degeneracy maps in \( {\left\{ {K}_{n}\coprod \left\{ {c}_{n}\right\} \right\} }_{n \geq 0} \) to the sequence of sets \[ {\left( CK\right... | It follows now from Lemma 17.1 that \( \left| {CK}\right| \cong \left| K\right| \times I/\left| K\right| \times \{ 1\} \) . In particular, - for any simplicial set \( K \) the realization \( \left| {CK}\right| \) of the cone on \( K \) is a contractible \( {CW} \) complex. | Yes |
If \( K \) is an extendable simplicial set then \( \left| K\right| \) is contractible. | Indeed, extendable simplicial sets \( K \) have the property that if \( E \) is sub simplicial set of a simplicial set \( L \) then any morphism \( \varphi : E \rightarrow K \) extends to a morphism \( L \rightarrow K \) . (This was proved in Proposition 10.4(ii) for simplicial cochain complexes, but the argument appli... | Yes |
Proposition 17.6 If \( \varrho : K \rightarrow L \) is a simplicial fibre bundle with fibre \( F \) then \( \left| \varrho \right| : \left| K\right| \rightarrow \left| L\right| \) is a fibre bundle with fibre \( \left| F\right| \) . | proof: Recall from \( §{17}\left( \mathrm{a}\right) \) that the cells of \( \left| L\right| \) are the non-degenerate simplices \( \sigma \) of \( L \) and that the characteristic maps are the maps \( {q}_{\sigma } : \{ \sigma \} \times {\Delta }^{n} \rightarrow \left| L\right| \) . It is immediate from the definitions... | Yes |
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