Q
stringlengths 4
3.96k
| A
stringlengths 1
3k
| Result
stringclasses 4
values |
|---|---|---|
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 \( {\Delta }^{n} \) and let \( L \) be the union of the other faces. Then \( L \) is contractible to \( {e}_{n} \) . Hence \( {H}_{ * }\left( {L;\mathbb{k}}\right) \overset{ \cong }{ \rightarrow }{H}_{ * }\left( {{\Delta }^{n};\mathbb{k}}\right) \), and the long exact sequence for \( \phi \subset L \subset {\Delta }^{n} \) implies that \( {H}_{ * }\left( {{\Delta }^{n}, L;\mathbb{R}}\right) = 0 \) . Thus the long exact sequence for \( L \subset \partial {\Delta }^{n} \subset {\Delta }^{n} \) provides an isomorphism \[ \partial : {H}_{ * }\left( {{\Delta }^{n},\partial {\Delta }^{n};\mathbb{k}}\right) \overset{ \cong }{ \rightarrow }{H}_{* - 1}\left( {\partial {\Delta }^{n}, L;\mathbb{k}}\right) . \] On the other hand, shrinking the last coordinate shows that the inclusion \( \left( {{\Delta }^{n - 1},\partial {\bigtriangleup }^{n - 1}}\right) \rightarrow \left( {\partial {\bigtriangleup }^{n} - \left\{ {e}_{n}\right\}, L - \left\{ {e}_{n}\right\} }\right) \) is a homotopy equivalence. This yields \[ {H}_{ * }\left( {\partial {\bigtriangleup }^{n}, L;\mathbf{k}}\right) \xleftarrow[\text{excision }]{ \cong }{H}_{ * }\left( {\partial {\bigtriangleup }^{n} - \left\{ {e}_{n}\right\}, L - \left\{ {e}_{n}\right\} ;\mathbf{k}}\right) \xleftarrow[]{ \cong }{H}_{ * }\left( {{\Delta }^{n - 1},\partial {\bigtriangleup }^{n - 1};\mathbf{k}}\right) \] Combined with the isomorphism above this gives an isomorphism \[ {H}_{ * }\left( {{\Delta }^{n},\partial {\Delta }^{n};\mathbb{R}}\right) \cong {H}_{* - 1}\left( {{\Delta }^{n - 1},\partial {\Delta }^{n - 1};\mathbb{R}}\right) \] which carries \( \left\lbrack {\bigtriangleup }^{n}\right\rbrack \) to \( {\left( -1\right) }^{n}\left\lbrack {\Delta }^{n - 1}\right\rbrack \) . The lemma follows by induction on \( n \) .
|
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 \( {\sigma }^{\prime } : {\bigtriangleup }^{n} \rightarrow X \) and a homotopy \( {\Phi }_{\sigma } : {\bigtriangleup }^{n} \times I \rightarrow Y \) from \( \sigma \) to \( \varphi \circ {\sigma }^{\prime } \) such that\n\n\[ \n{\sigma }^{\prime } \circ {\lambda }_{i} = {\left( \sigma \circ {\lambda }_{i}\right) }^{\prime }\;\text{ and }\;{\Phi }_{\sigma } \circ \left( {{\lambda }_{i} \times {id}}\right) = {\Phi }_{\sigma \circ {\lambda }_{i}},\;0 \leq i \leq n \n\]\n\n(4.16)\n\nIndeed we proceed by induction on \( n \) . Conditions (4.16) define \( {\sigma }^{\prime } \) on \( \partial {\bigtriangleup }^{n} \) and \( {\Phi }_{\sigma } \) on \( \partial {\bigtriangleup }^{n} \times I \) . Now the extension to \( {\sigma }^{\prime } \) and \( {\Phi }_{\sigma } \) is just the Whitehead lifting lemma 1.5.\n\nDefine a linear map \( f : C{S}_{ * }\left( {Y;\mathbb{R}}\right) \rightarrow C{S}_{ * }\left( {X;\mathbb{R}}\right) \) by \( \sigma \mapsto {\sigma }^{\prime } \) . The conditions above imply that \( f \) is a chain map and that \( C{S}_{ * }\left( \varphi \right) \circ f - {id} = {hd} + {dh} \), where \( h\left( \sigma \right) = C{S}_{ * }\left( {\Phi }_{\sigma }\right) \circ {EZ}\left( {\sigma \otimes i{d}_{I}}\right) . \n\nA second application of the Whitehead lemma gives, for each \( \tau : {\bigtriangleup }^{n} \rightarrow X \) , a homotopy \( {\Psi }_{\tau } : {\bigtriangleup }^{n} \times I \rightarrow X \) from \( {\left( \varphi \circ \tau \right) }^{\prime } \) to \( \tau \) such that \( {\Psi }_{\tau } \circ \left( {{\lambda }_{i} \times {id}}\right) = \) \( {\Psi }_{\tau \circ {\lambda }_{i}},0 \leq i \leq n \) . This then implies that \( f \circ C{S}_{ * }\left( \varphi \right) - {id} = {dk} + {kd} \) . Hence \( C{S}_{ * }\left( \varphi \right) \) is a chain equivalence and \( {C}_{ * }\left( \varphi \right) \) is a quasi-isomorphism.
|
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 cycle \( {z}_{\alpha } \in \) \( {C}_{n + 1}\left( {{X}_{n + 1},{X}_{n};\mathbb{k}}\right) \), and lift \( {z}_{\alpha } \) to a chain \( {w}_{\alpha } \in {C}_{n + 1}\left( {{X}_{n + 1};\mathbb{k}}\right) \) . Then \( d{w}_{\alpha } \) is an \( n \) -cycle in \( {C}_{ * }\left( {{X}_{n};\mathbb{k}}\right) \) . Since \( m\left( n\right) \) is a quasi-isomorphism there is a cycle \( {v}_{\alpha } \in {C}_{n} \) and a chain \( {a}_{\alpha } \in {C}_{n + 1}\left( {{X}_{n};\mathbb{R}}\right) \) such that \( m\left( n\right) \left( {v}_{\alpha }\right) = d{w}_{\alpha } + d{a}_{\alpha } \) . Since \( H\left( {\widetilde{m}\left( n\right) }\right) \) is the identity, it is immediate that \( {v}_{\alpha } = \partial {c}_{\alpha } \) . Extend \( m\left( n\right) \) to a morphism \( m\left( {n + 1}\right) \) by setting \( m\left( {n + 1}\right) \left( {c}_{\alpha }\right) = {w}_{\alpha } + {a}_{\alpha } \) . Since \( {w}_{\alpha } + {a}_{\alpha } \) also projects to \( {z}_{\alpha },\widetilde{m}\left( {n + 1}\right) \left( {c}_{\alpha }\right) = {z}_{\alpha } \) and \( H\left( {\widetilde{m}\left( {n + 1}\right) }\right) {c}_{\alpha } = \left\lbrack {z}_{\alpha }\right\rbrack = {c}_{\alpha } \), as desired.\n\nFinally, since \( H\left( {m\left( n\right) }\right) \) and \( H\left( {\widetilde{m}\left( {n + 1}\right) }\right) \) are isomorphisms so is \( H\left( {m\left( {n + 1}\right) }\right) \) by the five lemma (3.1). The sequence \( m\left( n\right) \) so constructed defines a cellular model \( m : \left( {{C}_{ * },\partial }\right) \rightarrow {C}_{ * }\left( {X, A;\mathbb{R}}\right) \) . Moreover for any cellular model we have, in the same way, that each \( m\left( n\right) \) is a quasi-isomorphism. Formula (4.17) identifies \( {H}_{i}\left( {m\left( n\right) }\right) \) with \( {H}_{i}\left( m\right) \) for \( i < n \) ; hence \( m \) is a quasi-isomorphism.
|
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 \( {\pi }_{n}\left( {{ \vee }_{\alpha }{S}_{\alpha }^{n}}\right) = {\bigoplus }_{\alpha }\mathbb{Z}{g}_{\alpha } \) . Let \( {f}_{\beta } : \left( {{S}_{\beta }^{n}, * }\right) \rightarrow { \vee }_{\alpha }{S}_{\alpha }^{n} \) represent \( {r}_{\beta } \) . Then the proof of the Hurewicz theorem also identifies \( {\pi }_{n}\left( {{ \vee }_{\alpha }{S}_{\alpha }^{n}{ \cup }_{\left\{ {f}_{\beta }\right\} }\mathop{\coprod }\limits_{\beta }{D}_{\beta }^{n + 1}}\right) = \pi \) . Put \( {Y}_{n + 1} = { \vee }_{\alpha }{S}_{\alpha }^{n}{ \cup }_{\left\{ {f}_{\beta }\right\} }\mathop{\coprod }\limits_{\beta }{D}_{\beta }^{n + 1} \) . Now create \( {Y}_{n + 2} \) by adding \( \left( {n + 2}\right) \) - cells to \( {Y}_{n + 1} \) to kill \( {\pi }_{n + 1}\left( {Y}_{n + 1}\right) \) and continue this process inductively, creating \( {Y}_{k + 1} \) by adding \( \left( {k + 1}\right) \) - cells to \( {Y}_{k} \) to kill \( {\pi }_{k}\left( {Y}_{k}\right) \) . The Cellular approximation theorem 1.1 will show that \( Y \) is a \( K\left( {\pi, n}\right) \) . For \( n = 1 \) we simply note that, since path spaces are contractible, the long exact homotopy sequence (2.2) for the path space fibration implies that \( {\Omega K}\left( {\pi ,2}\right) \) is a \( K\left( {\pi ,1}\right) \) . Let \( \left( {X,{x}_{0}}\right) \) be a cellular model for an Eilenberg-MacLane space \( \left( {Y,{y}_{0}}\right) \) and suppose \( K\left( {\pi, n}\right) \) is any Eilenberg-MacLane space of the same type. Proposition 4.20 gives a map \( g : \left( {X,{x}_{0}}\right) \rightarrow K\left( {\pi, n}\right) \) such that \( {\pi }_{n}\left( g\right) \) is the identity map of \( \pi \) . Thus \( \varphi \) is a weak homotopy equivalence.
|
Yes
|
Proposition 5.3 (i) If \( \mathbb{k} \) is a field then \( \alpha \) is always an isomorphism and so \( \langle \) , \( \bar{)\text{ }{is}\text{ }{non}\text{-}{degenerate}\text{. }{In}\text{ }{particular}\text{,}{H}^{ * }\left( {X, A;{Ik}}\right) \text{ }{hasfinite}\text{ }{type}\text{ }{if}\text{ }{and}\text{ }{only}\text{ }{if}\text{ }} \) \( {H}_{ * }\left( {X, A;\mathbb{k}}\right) \) does.
|
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\[ S{ \otimes }_{R}M\left( k\right) = S{ \otimes }_{R}M\left( {k - 1}\right) \oplus \left( {S \otimes Z\left( k\right) }\right), d : Z\left( k\right) \rightarrow S{ \otimes }_{R}M\left( {k - 1}\right) .\n\]\n\nThis exhibits \( \left\{ {S{ \otimes }_{R}M\left( k\right) }\right\} \) as a semifree filtration for \( \left( {S{ \otimes }_{R}M, d}\right) \) .
|
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 \( \\left( {R, d}\\right) \) -semifree. Then \( \\left( {M, d}\\right) \) itself is semifree.
|
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 \( Z\\left( {k,\\ell }\\right) \) a free graded \( \\mathbb{k} \) -module and\n\n\\[ \nd : Z\\left( {k,\\ell }\\right) \\rightarrow M\\left( {k - 1}\\right) \\oplus \\left( {R \\otimes \\left\\lbrack {{\\bigoplus }_{i < \\ell }Z\\left( {k, i}\\right) }\\right\\rbrack }\\right) . \n\\]\n\nThus \( M \) is \( R \) -free on the union \( \\left\\{ {z}_{\\alpha }\\right\\} \) of the bases of the \( \\mathbb{k} \) -modules \( Z\\left( {k,\\ell }\\right) \), with \( Z\\left( {k,\\ell }\\right) \) a free graded \( \\mathbb{I}k \) -module and\n\n\\[ \nd : Z\\left( {k,\\ell }\\right) \\rightarrow M\\left( {k - 1}\\right) \\oplus \\left( {R \\otimes \\left\\lbrack {{\\bigoplus }_{i < \\ell }Z\\left( {k, i}\\right) }\\right\\rbrack }\\right) . \n\\]\n\nDefine an increasing family \( W\\left( 0\\right) \\subset W\\left( 1\\right) \\subset \\cdots \) of free \( k \) -modules inductively as follows: \( W\\left( 0\\right) \) is spanned by the \( {z}_{\\alpha } \) for which \( d{z}_{\\alpha } = 0 \) and \( W\\left( m\\right) \) is spanned by the \( {z}_{\\alpha } \) for which \( d{z}_{\\alpha } \\in R \\cdot W\\left( {m - 1}\\right) \) . Then \( \\{ R \\cdot W\\left( m\\right) \\} \) will be a semifree filtration of \( \\left( {M, d}\\right) \), provided that each \( {z}_{\\alpha } \) is in some \( W\\left( m\\right) \) . Write \( \\left( {i, j}\\right) < \\left( {k,\\ell }\\right) \) if \( i < k \) or if \( i = k \) and \( j < \\ell \) . If \( {z}_{\\alpha } \\in Z\\left( {k,\\ell }\\right) \) then \( d{z}_{\\alpha } = \\sum {x}_{\\beta }{z}_{\\beta } \) with \( {x}_{\\beta } \\in R \) and \( {z}_{\\beta } \\in Z\\left( {i, j}\\right) \), some \( \\left( {i, j}\\right) < \\left( {k,\\ell }\\right) \) . We may assume by induction that each such \( {z}_{\\beta } \) is in some \( W\\left( {m}_{\\beta }\\right) \) . Put \( m = \\mathop{\\max }\\limits_{\\beta }{m}_{\\beta } \) . Then \( d{z}_{\\alpha } \\in W\\left( m\\right) \) and so \( {z}_{\\alpha } \\in W\\left( {m + 1}\\right) \) .
|
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 \( {f}^{\prime } \in {\operatorname{Hom}}_{R}\left( {M, P}\right) \) and \( {g}^{\prime } \in {\operatorname{Hom}}_{R}\left( {M, Q}\right) \) satisfying \( d\left( {f}^{\prime }\right) = f \) and \( d\left( {g}^{\prime }\right) = \eta \circ {f}^{\prime } - g \) . Let \( r = \deg f \) .\n\nChoose a semifree filtration \( \{ M\left( k\right) \} \) of \( M \), put \( M\left( {-1}\right) = 0 \) and, as in Remark 6.1 write \( M\left( k\right) = M\left( {k - 1}\right) \oplus \left( {R \otimes Z\left( k\right) }\right) \) where \( Z\left( k\right) \) is \( \mathbb{k} \) -free and \( d : Z\left( k\right) \rightarrow \) \( M\left( {k - 1}\right) \) . We construct \( {f}^{\prime } \) and \( {g}^{\prime } \) inductively by extending from \( M\left( {k - 1}\right) \) to \( M\left( k\right) \) .\n\nFor this let \( \left\{ {z}_{\alpha }\right\} \) be a basis of \( Z\left( k\right) \), put \( {p}_{\alpha } = f\left( {z}_{\alpha }\right) - {\left( -1\right) }^{r}{f}^{\prime }\left( {d{z}_{\alpha }}\right) \) and put \( {q}_{\alpha } = g\left( {z}_{\alpha }\right) - {\left( -1\right) }^{r}{g}^{\prime }\left( {d{z}_{\alpha }}\right) \) . By hypothesis the equations \( d\left( {f}^{\prime }\right) = f \) and \( d\left( {g}^{\prime }\right) = \eta \circ f - g \) are satisfied in \( M\left( {k - 1}\right) \) . It follows after a short calculation that \( d{p}_{\alpha } = 0 \) and \( \eta \left( {p}_{\alpha }\right) = d{q}_{\alpha } \) . Since \( \eta \) is a quasi-isomorphism there are (by Lemma 3.2) elements \( {p}_{\alpha }^{\prime } \in P \) and \( {q}_{\alpha }^{\prime } \in Q \) such that \( d{p}_{\alpha }^{\prime } = {p}_{\alpha } \) and \( d{q}_{\alpha }^{\prime } = \eta \left( {p}_{\alpha }^{\prime }\right) - {q}_{\alpha } \) . Now extend \( {f}^{\prime } \) and \( {g}^{\prime } \) to \( R \) -linear maps in \( M\left( k\right) \) by putting \( {f}^{\prime }\left( {z}_{\alpha }\right) = {p}_{\alpha }^{\prime } \) and \( {g}^{\prime }\left( {z}_{\alpha }\right) = {q}_{\alpha }^{\prime }.
|
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 \( \theta : M \rightarrow M \) and \( {\theta }^{\prime } : {M}^{\prime } \rightarrow {M}^{\prime } \) such that\n\n\[ \n{f}^{\prime } \circ f - i{d}_{M} = {d\theta } + {\theta d}\text{ and }f \circ {f}^{\prime } - i{d}_{{M}^{\prime }} = d{\theta }^{\prime } + {\theta }^{\prime }d. \n\]\n\n(6.8)\n\nApply \( {\operatorname{Hom}}_{R}\left( {-, i{d}_{{P}^{\prime }}}\right) \) to these formulae to conclude that \( {\operatorname{Hom}}_{R}\left( {f, i{d}_{{P}^{\prime }}}\right) \) is an equivalence. Since \( \left( {M, d}\right) \) is semifree, \( {\operatorname{Hom}}_{R}\left( {i{d}_{M}, g}\right) \) is just the quasi-isomorphism\n\n\( {\operatorname{Hom}}_{R}\left( {M, g}\right) \) of Proposition 6.4 (i).\n\nThus \( {\operatorname{Hom}}_{R}\left( {f, g}\right) = {\operatorname{Hom}}_{R}\left( {i{d}_{M}, g}\right) \circ {\operatorname{Hom}}_{R}\left( {f, i{d}_{{P}^{\prime }}}\right) \) is a quasi-isomorphism.
|
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 \( \varphi, h \) and \( f \) are quasi-isomorphisms, so is \( h{ \otimes }_{\varphi }f \) .
|
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\[ \varphi { \otimes }_{R}{id} : \left( {R{ \otimes }_{R}M, d}\right) \rightarrow \left( {S{ \otimes }_{R}M, d}\right) \]\n\nis a quasi-isomorphism, by Proposition 6.7 (ii). On the other hand, \( f \) extends to the morphism of \( \left( {S, d}\right) \) -modules\n\n\[ {f}^{\prime } : \left( {S{ \otimes }_{R}M, d}\right) \rightarrow \left( {{M}^{\prime }, d}\right) ,\;{f}^{\prime }\left( {y{ \otimes }_{R}m}\right) = {yf}\left( m\right) ,\;y \in S, m \in M. \]\n\nClearly \( f = {f}^{\prime } \circ \left( {\varphi { \otimes }_{R}{id}}\right) \) and so \( {f}^{\prime } \) is also a quasi-isomorphism.\n\n(i) Note that \( {\operatorname{Hom}}_{S}\left( {S{ \otimes }_{R}M, - }\right) = {\operatorname{Hom}}_{R}\left( {M, - }\right) \) and that \( {\operatorname{Hom}}_{\varphi }\left( {f, g}\right) \) is the composite of the morphisms\n\n\[ {\operatorname{Hom}}_{R}\left( {i{d}_{M}, g}\right) : {\operatorname{Hom}}_{R}\left( {M,{P}^{\prime }}\right) \rightarrow {\operatorname{Hom}}_{R}\left( {M, P}\right) \]\n\nand\n\n\[ {\operatorname{Hom}}_{S}\left( {{f}^{\prime }, i{d}_{{P}^{\prime }}}\right) : {\operatorname{Hom}}_{S}\left( {{M}^{\prime },{P}^{\prime }}\right) \rightarrow {\operatorname{Hom}}_{S}\left( {S{ \otimes }_{R}M,{P}^{\prime }}\right) .\n\nIn view of our remarks above, Proposition 6.7 (i) implies that both these morphisms are quasi-isomorphisms.\n\n(ii) Observe that \( {Q}^{\prime }{ \otimes }_{S}\left( {S{ \otimes }_{R}M}\right) = {Q}^{\prime }{ \otimes }_{R}M \) and that \( h{ \otimes }_{\varphi }f \) is the composite of\n\n\[ i{d}_{{Q}^{\prime }}{ \otimes }_{S}{f}^{\prime } : {Q}^{\prime }{ \otimes }_{S}\left( {S{ \otimes }_{R}M}\right) \rightarrow {Q}^{\prime }{ \otimes }_{S}{M}^{\prime } \]\n\nwith\n\n\[ h{ \otimes }_{R}{\operatorname{id}}_{M} : Q{ \otimes }_{R}M \rightarrow {Q}^{\prime }{ \otimes }_{R}M.\n\nNow apply Proposition 6.7 (ii).
|
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}^{n}\\right) }_{k} = \\left\\{ {\\begin{array}{ll} 0, & k < n - 1 \\\\ {\\left( \\operatorname{Im}d\\right) }_{k}, & k = n - 1 \\\\ {R}_{k}, & k \\geq n \\end{array}\\text{ and }{\\left( {J}^{n}\\right) }_{k} = \\left\\{ \\begin{array}{ll} 0, & k \\leq n - 1 \\\\ {\\left( \\ker d\\right) }_{k}, & k = n \\\\ {R}_{k}, & k > n. \\end{array}\\right. }\\right. \n\\]\n\nThese constructions are clearly natural.\n\nBy inspection, \\( H\\left( {{I}^{n}/{J}^{n}}\\right) = 0 \\) . Since \\( \\left( {N, d}\\right) \\) is semifree, \\( N{ \\otimes }_{R}\\{ 0\\} \\overset{ \\simeq }{ \\rightarrow }N{ \\otimes }_{R} \\) \\( {I}^{n}/{J}^{n} \\) . Thus\n\n\\[ \nN{ \\otimes }_{R}{I}^{n}/{J}^{n}\\overset{ \\simeq }{ \\rightarrow }{N}^{\\prime }{ \\otimes }_{R}{\\left( {I}^{\\prime }\\right) }^{n}/{\\left( {J}^{\\prime }\\right) }^{n} \n\\]\n\nsince both sides have zero homology.\n\nAlso by inspection,\n\n\\[ \n{\\left( {J}^{n}/{I}^{n + 1}\\right) }_{k} = \\left\\{ \\begin{array}{ll} {H}_{n}\\left( R\\right) , & k = n \\\\ 0, & \\text{ otherwise. } \\end{array}\\right. \n\\]\n\nThus we may write \\( N{ \\otimes }_{R}{J}^{n}/{I}^{n + 1} = \\left( {N{ \\otimes }_{R}{H}_{0}\\left( R\\right) }\\right) { \\otimes }_{{H}_{0}\\left( R\\right) }{H}_{n}\\left( R\\right) \\) . Now \\( N{ \\otimes }_{R}{H}_{0}\\left( R\\right) \\rightarrow {N}^{\\prime }{ \\otimes }_{{R}^{\\prime }}{H}_{0}\\left( {R}^{\\prime }\\right) \\) is a quasi-isomorphism of semifree \\( {H}_{0}\\left( R\\right) - \\) modules. Tensoring this with the isomorphism \\( {H}_{n}\\left( R\\right) \\overset{ \\cong }{ \\rightarrow }{H}_{n}\\left( {R}^{\\prime }\\right) \\) produces a quasi-isomorphism (Theorem 6.10) and so\n\n\\[ \nN{ \\otimes }_{R}{J}^{n}/{I}^{n + 1}\\overset{ \\simeq }{ \\rightarrow }{N}^{\\prime }{ \\otimes }_{R}{\\left( {J}^{\\prime }\\right) }^{n}/{\\left( {I}^{\\prime }\\right) }^{n + 1}. \n\\]\n\nNow an obvious induction via the five lemma shows that\n\n\\[ \nN{ \\otimes }_{R}R/{I}^{n}\\overset{ \\simeq }{ \\rightarrow }{N}^{\\prime }{ \\otimes }_{{R}^{\\prime }}{R}^{\\prime }/{\\left( {I}^{\\prime }\\right) }^{n},\\;n \\geq 1. \n\\]\n\nBut since \\( N = {N}_{ \\geq 0} \\), the modules \\( N \\) and \\( N{ \\otimes }_{R}R/{I}^{n + 3} \\) coincide in degrees \\( \\leq n + 1 \\) . Thus their homology coincides in degrees \\( \\leq n \\) . It follows that \\( H\\left( f\\right) \\) is an isomorphism.
|
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 \( {\mathcal{D}}_{C} \) and if \( {m}_{B} : {M}_{B}\overset{ \simeq }{ \rightarrow }E \) is a B-semifree resolution then\n\n\[{\bar{m}}_{B} : \mathbb{k}{ \otimes }_{B}{M}_{B} \rightarrow A\]\n\nis a quasi-isomorphism.
|
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 deduce that \( \bar{m} = \) \( \alpha \circ {\bar{m}}_{B} : \mathbb{k}{ \otimes }_{B}{M}_{B} \rightarrow A\left( 1\right) .\n\nOn the other hand, suppose \( m\left( 1\right) : M\left( 1\right) \overset{ \simeq }{ \rightarrow }E\left( 1\right) \) is any \( B\left( 1\right) \) -semifree resolution. Then, exactly as in the last part of the proof of Theorem 7.5, we can find a quasi-isomorphism \( \nu : M\left( 1\right) \rightarrow B\left( 1\right) { \otimes }_{B}{M}_{B} \) of \( B\left( 1\right) \) modules such that \( m \circ \nu \) is \( B\left( 1\right) \) -homotopic to \( m\left( 1\right) \) . Since \( \nu \) is a quasi-isomorphism so is \( \bar{\nu } : \mathbb{k}{ \otimes }_{B\left( 1\right) }M\left( 1\right) \rightarrow \mathbb{k}{ \otimes }_{B}{M}_{B} \) . Moreover, exactly as at the end of the proof of Theorem 7.5, \( H\left( \bar{m}\right) \circ H\left( \bar{\nu }\right) = H\left( \overline{m\left( 1\right) }\right) \) . Altogether we conclude that\n\n\( {\bar{m}}_{B} \) is a quasi-isomorphism \( \Leftrightarrow \bar{m} \) is a quasi-isomorphism\n\n\( \Leftrightarrow \overline{m\left( 1\right) } \) is a quasi-isomorphism.\n\nConsider the chain connecting \( \mathcal{D} \) to \( {\mathcal{D}}_{C} \), and recall that the semifree model for \( {\mathcal{D}}_{C} \) is just \( {m}_{Y} : {M}_{Y} \rightarrow {C}^{ * }\left( X\right) \) . Thus the argument above, repeated along the chain, shows that \( {\bar{m}}_{B} \) is a quasi-isomorphism if and only if \( {\bar{m}}_{Y} \) is. But \( {\bar{m}}_{Y} \) is a quasi-isomorphism by Theorem 7.5.
|
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\[ \nm : \left( {M, d}\right) \overset{ \simeq }{ \rightarrow }{C}_{ * }\left( P\right) \;\text{ and }\;{m}^{\prime } : \left( {{M}^{\prime }, d}\right) \overset{ \simeq }{ \rightarrow }{C}_{ * }\left( {P}^{\prime }\right) \n\]\n\nwith \( M \) and \( {M}^{\prime } \) concentrated in non-negative degrees. Since \( \left( {M, d}\right) \) is semifree and \( {m}^{\prime } \) is a quasi-isomorphism we can find\n\n\[ \nf : \left( {M, d}\right) \rightarrow \left( {{M}^{\prime }, d}\right) \;\text{ and }\;\theta : M \rightarrow {C}_{ * }\left( {P}^{\prime }\right) \n\]\n\nsuch that \( f \) is a morphism of \( {C}_{ * }\left( G\right) \) -modules, \( \theta \) is \( {C}_{ * }\left( G\right) \) -linear and \( {m}^{\prime }f = \) \( {C}_{ * }\left( \varphi \right) m + {d\theta } + {\theta d} \) (Proposition 6.4 (ii)).\n\nRecall the notation of Theorem 6.12, which we wish to apply here, noting that \( {H}_{0}\left( {{C}_{ * }\left( G\right) }\right) = {H}_{0}\left( {{C}_{ * }\left( {G}^{\prime }\right) }\right) = \mathbb{k} \) . Diagram (8.2) and its analogue for the fibration \( {p}^{\prime } \) yield the following diagram of chain complex morphisms: \n\nThis diagram may not commute. However, because \( \theta \) is \( {C}_{ * }\left( G\right) \) -linear it factors to give a linear map \( \bar{\theta } : M{ \otimes }_{{C}_{ * }\left( G\right) }\mathbb{k} \rightarrow {C}_{ * }\left( {X}^{\prime }\right) \) and \( {\bar{m}}^{\prime }\left( {f{ \otimes }_{{C}_{ * }\left( \alpha \right) }\mathbb{k}}\right) = {C}_{ * }\left( \psi \right) \bar{m} +\n\n\( d\bar{\theta } + \bar{\theta }d \) . Thus \( {C}_{ * }\left( \varphi \right) \) is a quasi-isomorphism if and only if \( f \) is and \( {C}_{ * }\left( \psi \right) \) is a quasi-isomorphism if and only if \( f{ \otimes }_{{C}_{ * }\left( \alpha \right) }\mathbb{R} \) is. Now apply Theorem 6.12.
|
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\[ \n{\pi }_{r}\left( X\right) { \otimes }_{\mathbb{Z}}\mathbb{k} = {H}_{r}\left( {X;\mathbb{k}}\right) \n\]
|
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 \( {\pi }_{r}\left( X\right) { \otimes }_{\mathbb{Z}}\mathbb{k}\overset{ \cong }{ \rightarrow }{H}_{r}\left( {X;\mathbb{Z}}\right) { \otimes }_{\mathbb{Z}}\mathbb{k} \) . On the other hand the identification \( {C}_{ * }\left( {X;\mathbb{Z}}\right) { \otimes }_{\mathbb{Z}}\mathbb{R} = {C}_{ * }\left( {X;\mathbb{R}}\right) \) defines a map \( {H}_{ * }\left( {X;\mathbb{Z}}\right) { \otimes }_{\mathbb{Z}}\mathbb{k} \rightarrow {H}_{ * }\left( {X;\mathbb{k}}\right) \) . Because \( \mathbb{k} \subset \mathbb{Q} \) the operation \( - { \otimes }_{\mathbb{Z}}\mathbb{k} \) preserves exactness, and it follows that this map too is an isomorphism.
|
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\[ \n{H}_{ * }\left( {X, A;\mathbb{Z}}\right) \text{ is }\mathcal{P} - \text{ local }\; \Leftrightarrow \;{H}_{ * }\left( {X, A;{\mathbb{F}}_{p}}\right) = 0, p \notin \mathcal{P}.\n\]\n\nIn particular\n\n\[ \n{H}_{ * }\left( {X,{pt};\mathbb{Z}}\right) \text{ is }\mathcal{P}\text{-local } \Leftrightarrow \;{H}_{ * }\left( {X;{\mathbb{F}}_{p}}\right) = {H}_{ * }\left( {{pt};{\mathbb{F}}_{p}}\right), p \notin \mathcal{P}.\n\]
|
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 that multiplication by \( p \) in \( {H}_{ * }\left( {X, A;\mathbb{Z}}\right) \) is an isomorphism precisely when \( {H}_{ * }\left( {X, A;{\mathbb{F}}_{p}}\right) = 0 \) .
|
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 case \( X \) has the weak homotopy type of a finite product of spaces \( K\left( {{\Gamma }_{i},1}\right) \) with \( {\Gamma }_{i} \) a cyclic \( \mathbb{R} \) -module. Exactly as in the proof of Case 3 in (8.6) we are reduced to the case \( \pi \) is a cyclic \( k \) -module.\n\nIf \( \pi \cong \mathbb{k} \) then \( X \) has the weak homotopy type of \( {S}_{\mathcal{P}}^{1} \) and \( {H}_{ * }\left( {X,{pt};\mathbb{Z}}\right) \) is \( {\mathcal{P}}_{ - } \) local by the calculation above of the homology of \( {S}_{\mathcal{P}}^{1} \) . If \( \pi ≆ \mathbb{k} \) then there is a short exact sequence of \( \mathbb{k} \) -modules of the form\n\n\[ 0 \rightarrow \mathbb{k} \rightarrow \mathbb{k} \rightarrow \pi \rightarrow 0 \]\n\nExactly as in Case 1 of Proposition 8.8, this leads to a morphism of \( {\Omega F} \) -fibrations of the form\n\n\n\nin which both \( {\Omega F} \) and \( {\Omega E} \) are Eilenberg-MacLane spaces of type \( \left( {\mathbb{R},1}\right) \) . Thus as observed above both \( {\Omega F} \) and \( {\Omega E} \) have the weak homotopy type of \( {S}_{\mathcal{P}}^{1} \) . Hence \( {H}_{ * }\left( {{\Omega F};{\mathbb{F}}_{p}}\right) = {H}_{ * }\left( {{\Omega E};{\mathbb{F}}_{p}}\right) = {H}_{ * }\left( {{pt},{\mathbb{F}}_{p}}\right) \) . Thus Theorem 8.5 asserts that \( {H}_{ * }\left( {{pt};{\mathbb{F}}_{p}}\right) \overset{ \cong }{ \rightarrow }{H}_{ * }\left( {{\Omega K}\left( {\pi ,2}\right) ;{\mathbb{F}}_{p}}\right) \) . Since \( {\Omega K}\left( {\pi ,2}\right) \) has the weak homotopy type of \( X \), the Lemma holds for \( X \) .\n\nThis establishes Lemma 9.4 for Eilenberg-MacLane spaces of type \( \left( {\pi ,1}\right) \) . The case when \( X \) has type \( \left( {\pi, n}\right), n > 1 \), follows by induction on \( n \) ; apply Theorem 8.5 to the morphism of \( {\Omega X} \) -fibrations\n\n
|
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 \( {\pi }_{ * }\left( \varphi \right) { \otimes }_{\mathbb{Z}}\mathbb{k} \) . Thus Theorem 9.6 is a special case of the Whitehead-Serre Theorem 8.6.
|
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 \( \Delta \left\lbrack n\right\rbrack \) .
|
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 bijection.
|
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 simplicial sets \( K \),\n\n\[ H\left( {\theta \left( K\right) }\right) : H\left( {D\left( K\right) }\right) \rightarrow H\left( {E\left( K\right) }\right) \]\n\nis an isomorphism.
|
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 third. Moreover Proposition 10.4 (i), identifies \( \theta \left( {\Delta \left\lbrack n\right\rbrack }\right) : D\left( {\Delta \left\lbrack n\right\rbrack }\right) \rightarrow E\left( {\Delta \left\lbrack n\right\rbrack }\right) \) with \( {\theta }_{n} \) : \( {D}_{n} \rightarrow {E}_{n} \) . Hence \( \theta \left( {\Delta \left\lbrack n\right\rbrack }\right) \) is a quasi-isomorphism, \( n \geq 0 \) . We now use induction on \( n \) to show that for any simplicial set \( K,\theta \left( {K\left( n\right) }\right) \) is a quasi-isomorphism.\n\nIndeed, this is vacuous for \( n = - 1 \) . Suppose it holds for some \( n - 1 \) . From Lemma 10.6 and the remarks above it follows that \( \theta \left( -\right) \) is a quasi-isomorphism when \( \left( -\right) \) is in turn given by:\n\n\[ \partial \Delta \left\lbrack n\right\rbrack ,\;\left( {\Delta \left\lbrack n\right\rbrack ,\partial \Delta \left\lbrack n\right\rbrack }\right) ,\left( {K\left( n\right), K\left( {n - 1}\right) }\right) \text{ and }K\left( n\right) . \]\n\nThis completes the inductive step. Along the way we have also shown that each \( \theta \left( {K\left( n\right), K\left( {n - 1}\right) }\right) \) is a quasi-isomorphism.\n\nFinally,
|
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 \( {\left\{ {\Psi }_{\sigma } \in A\left( \Delta \left\lbrack n\right\rbrack ,\Delta \left\lbrack n - 1\right\rbrack \right) \right\} }_{\sigma \in N{K}_{n}} \) , we recall first that (Proposition 10.4 (i))\n\n\[ A\left( {\Delta \left\lbrack n\right\rbrack ,\Delta \left\lbrack {n - 1}\right\rbrack }\right) \subset A\left( {\Delta \left\lbrack n\right\rbrack }\right) = {A}_{n}. \]\n\nThis identifies the \( {\Psi }_{\sigma } \) as elements of \( {A}_{n} \) satisfying \( {\partial }_{i}{\Psi }_{\sigma } = 0,0 \leq i \leq n \) . Define \( \Phi \in A\left( {K\left( n\right), K\left( {n - 1}\right) }\right) \) by the three conditions:\n\n\[ \left\{ \begin{array}{lll} {\Phi }_{\sigma } & = 0 & ,\;\sigma \in {K}_{m}, m < n \\ {\Phi }_{\sigma } & = {\Psi }_{\sigma } & ,\;\sigma \in N{K}_{n} \\ {\Phi }_{{s}_{j}\sigma } & = {s}_{j}{\Phi }_{\sigma } & ,\;\text{ all }j,\sigma . \end{array}\right. \]\n\nClearly \( {\alpha \Phi } = \left\{ {\Psi }_{\sigma }\right\} \) .
|
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\[ {C}_{PL}\left( K\right) \rightarrow \left( {{C}_{PL} \otimes {A}_{PL}}\right) \left( K\right) \leftarrow {A}_{PL}\left( K\right) \]\n\nare quasi-isomorphisms.
|
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 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 \( \Delta \left\lbrack p\right\rbrack \) . It follows, by a straightforward calculation from the definitions and (10.2), that the correspondence \( \gamma \mapsto f \) is a cochain algebra morphism. To show it is injective, assume \( \gamma \mapsto 0 \) . The elements \( \alpha \in \Delta {\left\lbrack n\right\rbrack }_{p} \) are the linear simplices of the form \( \left\langle {{e}_{{i}_{0}},\ldots ,{e}_{{i}_{p}}}\right\rangle : {\Delta }^{p} \rightarrow {\Delta }^{n} \) with \( {i}_{0} \leq \ldots \leq {i}_{p} \), and these can all be written as composites of face and degeneracy maps. It follows that for any \( n \geq 0 \) and any \( \tau \in {K}_{n} \),\n\n\[ \left( {\gamma }_{\tau }\right) \left( \alpha \right) = \left( {\gamma }_{\tau }\right) \left( {\alpha \circ {c}_{p}}\right) = {\gamma }_{\tau \circ \alpha }\left( {c}_{p}\right) = 0. \]\n\nHence \( \gamma = 0 \) and our morphism is injective.\n\nOn the other hand, by Lemma 10.3 any \( \sigma \in {K}_{n} \) determines a unique simplicial map \( {\sigma }_{ * } : \Delta \left\lbrack n\right\rbrack \rightarrow K \) such that \( {\sigma }_{ * }\left( {c}_{n}\right) = \sigma \) . Thus, if \( f \in {C}^{p}\left( K\right) \), we may define \( \gamma \in {C}_{PL}^{p}\left( K\right) \) by \( {\gamma }_{\sigma } = {C}^{p}\left( {\sigma }_{ * }\right) \left( f\right) \) . Clearly \( \gamma \mapsto f \) and our morphism is surjective.
|
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 \( \Delta \left\lbrack p\right\rbrack \) . It follows, by a straightforward calculation from the definitions and (10.2), that the correspondence \( \gamma \mapsto f \) is a cochain algebra morphism. To show it is injective, assume \( \gamma \mapsto 0 \) . The elements \( \alpha \in \Delta {\left\lbrack n\right\rbrack }_{p} \) are the linear simplices of the form \( \left\langle {{e}_{{i}_{0}},\ldots ,{e}_{{i}_{p}}}\right\rangle : {\Delta }^{p} \rightarrow {\Delta }^{n} \) with \( {i}_{0} \leq \ldots \leq {i}_{p} \), and these can all be written as composites of face and degeneracy maps. It follows that for any \( n \geq 0 \) and any \( \tau \in {K}_{n} \),\n\n\[ \left( {\gamma }_{\tau }\right) \left( \alpha \right) = \left( {\gamma }_{\tau }\right) \left( {\alpha \circ {c}_{p}}\right) = {\gamma }_{\tau \circ \alpha }\left( {c}_{p}\right) = 0. \]\n\nHence \( \gamma = 0 \) and our morphism is injective.\n\nOn the other hand, by Lemma 10.3 any \( \sigma \in {K}_{n} \) determines a unique simplicial map \( {\sigma }_{ * } : \Delta \left\lbrack n\right\rbrack \rightarrow K \) such that \( {\sigma }_{ * }\left( {c}_{n}\right) = \sigma \) . Thus, if \( f \in {C}^{p}\left( K\right) \), we may define \( \gamma \in {C}_{PL}^{p}\left( K\right) \) by \( {\gamma }_{\sigma } = {C}^{p}\left( {\sigma }_{ * }\right) \left( f\right) \) . Clearly \( \gamma \mapsto f \) and our morphism is surjective.
|
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 10.12) the assertion (ii) of the theorem follows from assertion (i).\n\nTo prove (i) we show first that each \( {\oint }_{n} \) commutes with the differentials, which is essentially Stokes' theorem. For this it is sufficient to verify that\n\n\[ \n{\left( -1\right) }^{\frac{n\left( {n - 1}\right) }{2}}{\int }_{n}{d\Phi } = {\left( -1\right) }^{\frac{\left( {n - 1}\right) \left( {n - 2}\right) }{2}}\mathop{\sum }\limits_{{i = 0}}^{n}{\left( -1\right) }^{n + i + 1}{\int }_{n - 1}{\sigma }_{i}^{ * }\Phi ,\;\Phi \in {\left( {A}_{PL}\right) }_{n}^{n - 1},\n\]\n\nwhere \( {\sigma }_{i} = \left\langle {{e}_{0}\cdots {e}_{i - 1}{e}_{i + 1}\cdots {e}_{n}}\right\rangle : {\Delta }^{n - 1} \rightarrow {\Delta }^{n} \). By linearity it is enough to consider \( \Phi \) of the form \( {t}_{1}^{{k}_{1}}\cdots {t}_{n}^{{k}_{n}}d{t}_{1} \land \cdots \widehat{d{t}_{j}}\cdots \land d{t}_{n}\left( {d{t}_{j}}\right. \) is deleted). But in this case the formula is a straightforward calculation via (10.14).\n\nNext the compatibility of the \( {\oint }_{n} \) with the face and degeneracy morphisms follows from the equation\n\n\[ \n\left( {{\oint }_{r}{A}_{PL}\left( \sigma \right) \Phi }\right) \left( \tau \right) = {\left( -1\right) }^{\frac{k\left( {k - 1}\right) }{2}}{\int }_{k}{\tau }^{ * }{\sigma }^{ * }\Phi = {\left( -1\right) }^{\frac{k\left( {k - 1}\right) }{2}}{\int }_{k}{\left( \sigma \tau \right) }^{ * }\Phi\n\]\n\n\[ \n= \left( {{\oint }_{n}\Phi }\right) \left( {\sigma \tau }\right) = \left( {{C}_{PL}\left( \sigma \right) {\oint }_{n}\Phi }\right) \left( \tau \right) ,\n\]\n\nvalid for all simplicial maps \( {\Delta }^{k}\overset{\tau }{ \rightarrow }{\Delta }^{r}\overset{\sigma }{ \rightarrow }{\Delta }^{n} \) and all \( \Phi \in {\left( {A}_{PL}\right) }_{n}^{k} \).\n\nFinally, to show that the \( {\oint }_{n} \) are quasi-isomorphisms notice that \( {\oint }_{n}\left( 1\right) = 1 \) and recall that \( H\left( {\left( {A}_{PL}\right) }_{n}\right) = \mathbb{k} = H\left( {\left( {C}_{PL}\right) }_{n}\right), n \geq 0 \) (Lemmas 10.7 and 10.12).
|
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) \( {\gamma }_{M} \) is a quasi-isomorphism. The inclusion \( {S}_{ * }^{\infty }\left( M\right) \rightarrow {S}_{ * }\left( M\right) \) induces \( {\varrho }_{M} : {C}^{ * }\left( M\right) \rightarrow {C}_{\infty }^{ * }\left( M\right) \), and Theorem 10.9 identifies \( H\left( {\gamma }_{M}\right) \) and \( H\left( {\varrho }_{M}\right) \).\n\nOn the other hand if \( f, g : M \rightarrow N \) are smoothly homotopic then the homotopy \( {C}_{ * }\left( f\right) - {C}_{ * }\left( g\right) = {dh} + {hd} \) defined in \( §4\left( \mathrm{a}\right) \) restricts to a homotopy between \( {C}_{ * }^{\infty }\left( f\right) \) and \( {C}_{ * }^{\infty }\left( g\right) \) . Thus since the identity and constant maps in \( {\mathbb{R}}^{n} \) are smoothly homotopic we conclude that \( H\left( {{C}^{ * }\left( {\mathbb{R}}^{n}\right) }\right) = \mathbb{R} = H\left( {{C}_{\infty }^{ * }\left( {\mathbb{R}}^{n}\right) }\right) \) .\n\nNext, if \( U \) and \( V \) are open subsets of \( M \) then the barycentric subdivision argument [121] that shows that \( {C}_{ * }\left( U\right) + {C}_{ * }\left( V\right) \)
|
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 \( O = \mathop{\coprod }\limits_{i}{O}_{i} \) , \( W = \mathop{\coprod }\limits_{j}{W}_{j} \) and each \( {O}_{i} \) and \( {W}_{j} \) is a finite union of elements of the \( i \) -basis. If each \( {\theta }_{{V}_{\lambda }} \) is a quasi-isomorphism it follows by induction on \( p \) that each \( {\theta }_{{V}_{{\lambda }_{1}} \cup \cdots \cup {V}_{{\lambda }_{p}}} \) is a quasi-isomorphism and hence that \( {\theta }_{0},{\theta }_{W} \) and \( {\theta }_{O \cap W} \) are too. Thus in this case \( {\theta }_{M} \) is a quasi-isomorphism.\n\nNow suppose \( U \) is open in \( {\mathbb{R}}^{n} \) . A standard cube in \( {\mathbb{R}}^{n} \) is an open set \( V \) of the form \( \left( {{a}_{1},{b}_{1}}\right) \times \cdots \times \left( {{a}_{n},{b}_{n}}\right) \) and each \( {\theta }_{V} \) is a quasi-isomorphism by (i) above. But the standard cubes contained in \( U \) are an \( i \) -basis for \( U \) and so \( {\theta }_{U} \) is a quasi-isomorphism. But the open subsets of \( M \) diffeomorphic to open subsets of \( {\mathbb{R}}^{n} \) are an \( i \) -basis for \( M \) and so \( {\theta }_{M} \) is a quasi-isomorphism.
|
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 that \( \\left\\langle {\\omega ,\\left\\lbrack {S}^{k}\\right\\rbrack }\\right\\rangle = 1 \), and \( 1,\\omega \) is a basis for \( H\\left( {{A}_{PL}\\left( {S}^{k}\\right) }\\right) \). Let \( \\Phi \) be a representing cocycle for \( \\omega \). Now if \( k \) is odd then a minimal Sullivan model for \( {S}^{k} \) is given by \[ m : \\left( {\\Lambda \\left( e\\right) ,0}\\right) \\overset{ \\simeq }{ \\rightarrow }{A}_{PL}\\left( {S}^{k}\\right) ,\\;\\begin{array}{l} \\deg e = k. \\\\ {me} = \\Phi . \\end{array} \] Indeed, since \( k \) is odd,1 and \( e \) are a basis for the exterior algebra \( \\Lambda \\left( e\\right) \).
|
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 \( 1, e,{e}^{2},{e}^{3},\\ldots \) and this morphism is not a quasi-isomorphism. However, \( {\\Phi }^{2} \) is certainly a coboundary. Write \( {\\Phi }^{2} = {d\\Psi } \) and extend \( m \) to \[ m : \\left( {\\Lambda \\left( {e,{e}^{\\prime }}\\right), d}\\right) \\rightarrow {A}_{PL}\\left( {S}^{k}\\right) \] by setting \( \\deg {e}^{\\prime } = {2k} - 1, d{e}^{\\prime } = {e}^{2} \) and \( m{e}^{\\prime } = \\Psi \). A simple computation shows that \( 1, e \) represents a basis of \( H\\left( {\\Lambda \\left( {e,{e}^{\\prime }}\\right), d}\\right) \). Thus this is a minimal model for \( {S}^{k} \). Finally, observe that quasi-isomorphisms \[ \\left( {{\\Lambda e},0}\\right) \\rightarrow \\left( {{H}^{ * }\\left( {S}^{k}\\right) ,0}\\right), k\\text{ odd }\\text{ and }\\;\\left( {\\Lambda \\left( {e,{e}^{\\prime }}\\right), d}\\right) \\overset{ \\simeq }{ \\rightarrow }\\left( {{H}^{ * }\\left( {S}^{k}\\right) ,0}\\right), k\\text{ even } \] are given by \( e \\mapsto \\omega ,{e}^{\\prime } \\mapsto 0 \).
|
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 spaces has finite type. Let \( {p}^{X} : X \times Y \rightarrow X \) and \( {p}^{Y} : X \times Y \rightarrow Y \) be the projections. Then \( {A}_{PL}\left( {p}^{X}\right) \cdot {A}_{PL}\left( {p}^{Y}\right) : {A}_{PL}\left( X\right) \otimes \) \( {A}_{PL}\left( Y\right) \rightarrow {A}_{PL}\left( {X \times Y}\right) \) is a quasi-isomorphism of cochain algebras.
|
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 the induced map of cohomology with the map\n\n\[ \n{H}^{ * }\left( {X;\mathbb{R}}\right) \otimes {H}^{ * }\left( {Y;\mathbb{R}}\right) \rightarrow {H}^{ * }\left( {X \times Y;\mathbb{R}}\right)\n\]\n\ngiven by \( \alpha \otimes \beta \mapsto {H}^{ * }\left( {p}^{X}\right) \alpha \cup {H}^{ * }\left( {p}^{Y}\right) \beta \) . But Proposition 5.3(ii) asserts that this map is an isomorphism.\n\nSince \( {A}_{PL}\left( {p}^{X}\right) \cdot {A}_{PL}\left( {p}^{Y}\right) \) is a quasi-isomorphism so is\n\n\[ \n{m}_{X} \cdot {m}_{Y} : \left( {{\Lambda V}, d}\right) \otimes \left( {{\Lambda W}, d}\right) \overset{ \simeq }{ \rightarrow }{A}_{PL}\left( {X \times Y}\right) ,\n\]\n\nwhere \( \left( {{m}_{X} \cdot {m}_{Y}}\right) \left( {a \otimes b}\right) = {A}_{PL}\left( {p}^{X}\right) {m}_{X}a \cdot {A}_{PL}\left( {p}^{Y}\right) b \) . This exhibits \( \left( {{\Lambda V}, d}\right) \otimes \) \( \left( {{\Lambda W}, d}\right) \) as a Sullivan model for \( X \times Y \) . Observe that if \( \left( {{\Lambda V}, d}\right) \) and \( \left( {{\Lambda W}, d}\right) \) are minimal models then so is their tensor product.
|
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. (If it were, it would have to have a cocycle of degree 1). The cocycles 1 and \( {v}_{1}{v}_{2}{v}_{3} \) represent a basis for \( H\left( A\right) \), and so it has a minimal model\n\n\[ m : \left( {\Lambda \left( w\right) ,0}\right) \overset{ \simeq }{ \rightarrow }\left( {{\Lambda V}, d}\right) ,\;\deg w = 3, m\left( w\right) = {v}_{1}{v}_{2}{v}_{3}. \]
|
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 \( \left( {{\Lambda V}, d}\right) \) is minimal. It remains to show \( m \) is a quasi-isomorphism, and this follows at once from the assertion,\n\n\[ {H}^{i}\left( {m}_{k}\right) \text{ is }\left\{ {\begin{array}{l} \text{ an isomorphism for }i \leq k \\ \text{ injective for }i = k + 1 \end{array}\;, k \geq 2,}\right. \]\n\n(12.3)\n\nwhich we prove by induction on \( k \) .\n\nFor \( k = 2\left( {12.3}\right) \) was observed at the start of the construction above. Suppose (12.3) holds for some \( k \) . Since \( {m}_{k + 1} \) extends \( {m}_{k},\operatorname{Im}H\left( {m}_{k}\right) \subset \operatorname{Im}H\left( {m}_{k + 1}\right) \) . Thus \( {H}^{i}\left( {m}_{k + 1}\right) \) is surjective for \( i \leq k \) by induction and surjective for \( i = k + 1 \) by construction.\n\nTo show \( {H}^{i}\left( {m}_{k + 1}\right) \) is injective for \( i \leq k + 2 \) let \( \left\lbrack z\right\rbrack \) be a cohomology class in \( \ker {H}^{i}\left( {m}_{k + 1}\right) \), some \( i \leq k + 2 \) . We have to show \( \left\lbrack z\right\rbrack = 0 \) . If \( \deg \left\lbrack z\right\rbrack \leq k \) or if \( \deg \left\lbrack z\right\rbrack = k + 2 \) then \( z \in \Lambda {V}^{ \leq k} \) and \( \left\lbrack z\right\rbrack \in \ker {H}^{i}\left( {m}_{k}\right) \) . Thus \( \left\lbrack z\right\rbrack = 0 \) by induction if \( \deg \left\lbrack z\right\rbrack \leq k \) and by construction if \( \deg \left\lbrack z\right\rbrack = k + 2 \) .\n\nSuppose \( \deg \left\lbrack z\right\rbrack = k + 1 \) . Then \( z = \sum {\lambda }_{\alpha }{v}_{\alpha }^{\prime } + \sum {\lambda }_{\beta }{v}_{\beta }^{\prime \prime } + w \), some \( w \in \Lambda {V}^{ \leq k} \), where we use the notation from the construction above. Since \( {dz} = 0,\sum {\lambda }_{\beta }{z}_{\beta } = - {dw} \) and \( \sum {\lambda }_{\beta }\left\lbrack {z}_{\beta }\right\rbrack = 0 \) in \( H\left( {\Lambda {V}^{ \leq k}}\right) \) . By construction, this implies that each \( {\lambda }_{\beta } = 0 \) . But then \( {dw} = 0 \) and \( \sum {\lambda }_{\alpha }\left\lbrack {a}_{\alpha }\right\rbrack = {H}^{k + 1}\left( {m}_{k}\right) \left\lbrack w\right\rbrack \) . Again by construction, each \( {\lambda }_{\alpha } = 0 \) . Hence \( z = w \) and so \( \left\lbrack z\right\rbrack \in \ker {H}^{k + 1}\left( {m}_{k}\right) \) . By induction, \( \left\lbrack z\right\rbrack = 0 \) .
|
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\[ m : \left( {{\Lambda V}, d}\right) \overset{ \simeq }{ \rightarrow }{A}_{PL}\left( X\right) \]\n\nin which \( V = {\left\{ {V}^{i}\right\} }_{i \geq 2} \) and each \( {V}^{i} \) is finite dimensional. In general, \( {\Lambda V} \) will not be finite dimensiona \( \overline{\mathrm{l}} \), as is already shown by the even spheres \( {S}^{2r} \) (cf. Example 1).\n\nThere is, however, a ’non-free’ finite dimensional commutative model for \( X \) , constructed as follows. Put\n\n\[ {n}_{X} = \max \left\{ {i \mid {H}^{i}\left( {X;\mathbb{k}}\right) \neq 0}\right\} .\n\nWrite \( {\left( \Lambda V\right) }^{{n}_{X}} = H \oplus {\left( \operatorname{Im}d\right) }^{{n}_{X}} \oplus C \), where \( H \oplus {\left( \operatorname{Im}d\right) }^{{n}_{X}} = {\left( \ker d\right) }^{{n}_{X}} \) ; thus \( H\overset{ \cong }{ \rightarrow }{H}^{{n}_{X}}\left( {{\Lambda V}, d}\right) \cong {H}^{{n}_{X}}\left( {X;\mathbb{k}}\right) \). Choose a graded subspace \( I \subset \left( {{\Lambda V}, d}\right) \) so that\n\n\[ \begin{array}{l} {I}^{k} = 0, k < {n}_{X} - 1 \\ {I}^{k} = {\left( \Lambda V\right) }^{k}, k > {n}_{X} \end{array}\;\text{ and }\;\begin{array}{l} {I}^{{n}_{X} - 1} \oplus {\left( \ker d\right) }^{{n}_{X} - 1} = {\left( \Lambda V\right) }^{{n}_{X} - 1} \\ {I}^{{n}_{X}} = {\left( \operatorname{Im}d\right) }^{{n}_{X}} \oplus C. \end{array} \]\n\nSince \( V = {\left\{ {V}^{i}\right\} }_{i > 2}, I \) is an ideal. It is immediate from the construction that \( I \) is preserved by \( d \) and that \( H\left( {I, d}\right) = 0 \) (because \( {H}^{i}\left( {{\Lambda V}, d}\right) = 0, i > {n}_{X} \) ). Thus the quotient map\n\n\[ \eta : \left( {{\Lambda V}, d}\right) \overset{ \simeq }{ \rightarrow }\left( {\left( {\Lambda V}\right) /I, d}\right) \]\n\nis a quasi-isomorphism, and so \( \left( {\left( {\Lambda V}\right) /I, d}\right) \) is a finite-dimensional commutative model for \( X \).\n\nNote that \( \left( {\Lambda V}\right) /I \) vanishes in degrees \( k > {n}_{X} \) and that \( {\left\lbrack \left( \Lambda V\right) /I\right\rbrack }^{{n}_{X}} = \) \( {H}^{{n}_{X}}\left( {\left( {\Lambda V}\right) /I}\right) \).
|
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 that \( {\alpha \delta } = \varepsilon = {\beta \gamma } \), and that all other products of basis elements in \( {H}^{ + } \) are zero.\n\nWe can now use the procedure above to construct a minimal model for the cochain algebra \( \left( {H,0}\right) \) . This will have the form \( m : \left( {{\Lambda V}, d}\right) \overset{ \simeq }{ \rightarrow }\left( {H,0}\right) \), beginning with\n\n\[ \n{v}_{2}\;d{v}_{2} = 0\;m{v}_{2} = \beta \]\n\n\[ \n{V}^{3} : \;{u}_{1}\;d{u}_{1} = {v}_{1}^{2}\;m{u}_{1} = 0 \]\n\n\[ \n\begin{array}{lll} {u}_{2} & d{u}_{2} = {v}_{1}{v}_{2} & m{u}_{2} = 0 \end{array} \]\n\n\[ \n{u}_{3}\;d{u}_{3} = {v}_{2}^{2}\;m{u}_{3} = 0 \]\n\nNote that necessarily\n\n\[ \nm\left( {{v}_{1}{u}_{2} - {v}_{2}{u}_{1}}\right) = 0 = m\left( {{v}_{2}{u}_{2} - {v}_{1}{u}_{3}}\right) . \]\n\nThus we need to add\n\n\[ \n{V}^{4} : \;{x}_{1}\;d{x}_{1} = {v}_{1}{u}_{2} - {v}_{2}{u}_{1}\;m{x}_{1} = 0 \]\n\n\[ \n{x}_{2}\;d{x}_{2} = {v}_{2}{u}_{2} - {v}_{1}{u}_{3}\;m{x}_{2} = 0, \]\n\nand\n\n\[ \n{V}^{5} : \;{y}_{1}\;d{y}_{1} = 0\;m{y}_{1} = \gamma \]\n\n\[ \n{y}_{2}\;d{y}_{2} = 0\;m{y}_{2} = \delta . \]\n\nThe process turns out (but we can not yet prove this) to continue without end.\n\nObserve that this provides two distinct minimal Sullivan algebras with the same cohomology algebra.
|
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 \( V\left( {k - 1}\right) \) and let \( {v}_{\alpha } \) be a basis of \( {V}_{k} \) . Then \( {\varphi d}{v}_{\alpha } \) is defined and \( d\left( {{\varphi d}{v}_{\alpha }}\right) = \varphi \left( {{d}^{2}{v}_{\alpha }}\right) = 0 \) . Furthermore,\n\n\[ \n{\eta \varphi d}{v}_{\alpha } = {\psi d}{v}_{\alpha } = {d\psi }{v}_{\alpha }\n\]\n\nSince \( \eta \) is a surjective quasi-isomorphism we can find \( {a}_{\alpha } \in A \) so that \( d{a}_{\alpha } = {\varphi d}{v}_{\alpha } \) and \( \eta {a}_{\alpha } = \psi {v}_{\alpha } \) . Extend \( \varphi \) by setting \( \varphi {v}_{\alpha } = {a}_{\alpha } \) .
|
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 \( E\left( U\right) = \bigotimes \Lambda \left( {{u}_{\alpha },\delta {u}_{\alpha }}\right) \) and, since \( \mathbb{k} \) is a field, homology commutes with tensor products (Proposition 3.3). \( ▱ \)
|
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 inclusions at the endpoints and by \( {p}^{X} : X \times I \rightarrow X \) and \( {p}^{I} : X \times I \rightarrow I \) the projections. Then\n\n\n\n\nis a commutative diagram of cochain algebra morphisms.\n\nSince \( H\left( {{A}_{PL}\left( {p}^{X}\right) }\right) = {H}^{ * }\left( {{p}^{X};\mathbb{R}}\right) \) is an isomorphism, \( {A}_{PL}\left( {p}^{X}\right) \cdot {A}_{PL}\left( {p}^{I}\right) \) is a quasi-isomorphism. We now ’make it surjective’. Let \( U \subset {A}_{PL}\left( {X \times I}\right) \) be the kernel of \( \left( {{A}_{PL}\left( {j}_{0}\right) ,{A}_{PL}\left( {j}_{1}\right) }\right) \) . The inclusion of \( U \) extends to a unique cochain algebra morphism\n\n\[ \varrho : \left( {E\left( U\right) ,\delta }\right) \rightarrow {A}_{PL}\left( {X \times I}\right) \]\n\nExtend the diagram above to the commutative diagram\n\n\[ {A}_{PL}\left( X\right) \otimes \Lambda \left( {t,{dt}}\right) \otimes \left( {E\left( U\right) ,\delta }\right) \]\n\nHere \( {A}_{PL}\left( {p}^{X}\right) \cdot {A}_{PL}\left( {p}^{I}\right) \cdot \varrho \) is, obviously, surjective, and it follows from Lemma 12.5 that it is a quasi-isomorphism too.\n\nLet \( H : X \times I \rightarrow Y \) be a homotopy from \( {f}_{0} \) to \( {f}_{1} \) . Use Lemma 12.4 to lift \( {A}_{PL}\left( H\right) \psi : \left( {{\Lambda V}, d}\right) \rightarrow {A}_{PL}\left( {X \times I}\right) \) through the surjective quasi-isomorphism \( {A}_{PL}\left( {p}^{X}\right) \cdot {A}_{PL}\left( {p}^{I}\right) \cdot \varrho \) . This produces a morphism\n\n\[ \Psi : \left( {{\Lambda V}, d}\right) \rightarrow {A}_{PL}\left( X\right) \otimes \Lambda \left( {t,{dt}}\right) \otimes E\left( U\right) . \]\n\nThen set \( \Phi = \left( {{id} \otimes {id} \otimes \varepsilon }\right) \Psi \) ; it is the desired homotopy from \( {A}_{PL}\left( {f}_{0}\right) \psi \) to \( {A}_{PL}\left( {f}_{1}\right) \psi \)
|
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 \( \varepsilon \left( V\right) = 0 \) . We observe that for any morphism \( \varphi : \left( {{\Lambda V}, d}\right) \rightarrow \) \( \left( {A,0}\right) \) ,\n\n\[ \varphi \sim \varepsilon \Leftrightarrow \varphi = \varepsilon \]\n\nIn fact, suppose \( \Phi : \left( {{\Lambda V}, d}\right) \rightarrow A \otimes \Lambda \left( {t,{dt}}\right) \) is a homotopy from \( \varphi \) to \( \varepsilon \) and write \( V = \mathop{\bigcup }\limits_{{k \geq - 1}}V\left( k\right) \) with \( V\left( {-1}\right) = 0 \) and \( d : V\left( k\right) \rightarrow {\Lambda }^{ \geq 2}V\left( {k - 1}\right) \) . Assume by induction that \( \Phi : V\left( {k - 1}\right) \rightarrow A \otimes {\Lambda t} \otimes {dt} \) . Since \( {dt} \land {dt} = 0 \) it follows that \( \Phi \left( {{\Lambda }^{ \geq 2}V\left( {k - 1}\right) }\right) = 0. \)\n\nChoose \( v \in V\left( k\right) \) . Then \( d\left( {\Phi v}\right) = \Phi \left( {dv}\right) = 0 \) . Hence \( {\Phi v} \in \left( {A \otimes 1}\right) \oplus \left( {A \otimes {\Lambda t} \otimes {dt}}\right) \) . Since \( 0 = {\varepsilon v} = \left( {{id} \otimes {\varepsilon }_{1}}\right) {\Phi v} \) it follows that \( {\Phi v} \in A \otimes {\Lambda t} \otimes {dt} \) . In other words, \( \operatorname{Im}\Phi \subset A \otimes {\Lambda t} \otimes {dt} \) and \( \varphi = \left( {{id} \otimes {\varepsilon }_{0}}\right) \Phi = \varepsilon \) .
|
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 \( \left( {{X}_{\alpha },{x}_{\alpha }}\right) \) be based CW complexes, so that \( \varepsilon : {A}_{PL}\left( {X}_{\alpha }\right) \rightarrow \mathbb{k} \) are augmented cochain algebras. Denote by \( {j}_{\alpha } : \left( {{X}_{\alpha },{x}_{\alpha }}\right) \rightarrow \left( {\mathop{\bigvee }\limits_{\alpha }{X}_{\alpha },\bar{x}}\right) \) the different inclusions into the wedge. Thus \( \Phi \mapsto \left\{ {{A}_{PL}\left( {j}_{\alpha }\right) \Phi }\right\} \) defines a morphism\n\n\[ {A}_{PL}\left( {\mathop{\bigvee }\limits_{\alpha }{X}_{\alpha }}\right) \rightarrow {\left( \mathop{\prod }\limits_{\alpha }\right) }_{Ik}\left( {{A}_{PL}\left( {X}_{\alpha }\right) }\right) \]\n\nto the fibre product over \( \mathbb{R} \) of the \( {A}_{PL}\left( {X}_{\alpha }\right) \) . This morphism, which is obviously surjective, induces the analogue\n\n\[ {H}^{ * }\left( {\mathop{\bigvee }\limits_{\alpha }{X}_{\alpha };\mathbb{R}}\right) \rightarrow {\left( \mathop{\prod }\limits_{\alpha }\right) }_{Ik}\left( {{H}^{ * }\left( {{X}_{\alpha },\mathbb{R}}\right) }\right) \]\n\nin cohomology and a cellular chains argument \( \left( {§4\left( \mathrm{e}\right) }\right) \) shows this is an isomorphism.
|
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, \( {\Phi a} = a \otimes {f}_{1}\left( t\right) + {xy} \otimes {f}_{2}\left( t\right) + z \otimes {f}_{3}\left( t\right) {dt} \) . Similarly, \( {\Phi u} = u \otimes {g}_{1}\left( t\right) + {xyz} \otimes {g}_{2}\left( t\right) + {za} \otimes {g}_{3}\left( t\right) \) . From \( {d\Phi a} = 0 \) deduce \( {f}_{1},{f}_{2} \in \mathbb{R} \) . If \( {\varphi }_{0}a = a \) it follows that \( {f}_{1} = 1 \) and \( {f}_{2} = 0 \) . In this case the equation \( {d\Phi u} = {\left( \Phi a\right) }^{2} \) implies that \( {g}_{1} = 1 \), and \( d{g}_{2} = 0 \) . Thus \( {g}_{2} \in \mathbb{R} \) and \( {\varphi }_{0}u - {\varphi }_{1}u = {\lambda za} \) . In particular, \( \varphi \) is not homotopic to the identity.
|
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\[ \n\begin{array}{l} {x}_{1},{x}_{2},{x}_{3}\;y\;z\;v\;{w}_{1},{w}_{2} \\ d{x}_{i} = 0\;{dy} = {x}_{2}{x}_{3}\;{dz} = 0\;{dv} = y{x}_{2}\;d{w}_{1} = {z}^{2}, d{w}_{2} = z{x}_{1}{x}_{2} \\ \end{array} \n\]\n\nDefine two automorphisms \( \varphi ,\psi : \left( {{\Lambda V}, d}\right) \rightarrow \left( {{\Lambda V}, d}\right) \) by requiring\n\n\( {\varphi z} = z + {x}_{1}{x}_{2},\;\varphi {w}_{1} = {w}_{1} + 2{w}_{2},\;\varphi = \) identity on the other generators,\n\nand\n\n\[ \n\psi {x}_{1} = {x}_{1} + {x}_{3},\;\psi {w}_{2} = {w}_{2} - {zy},\;\psi = \text{identity on the other generators.} \n\]\n\nThen a homotopy \( \Phi : \left( {{\Lambda V}, d}\right) \rightarrow \left( {{\Lambda V}, d}\right) \otimes \Lambda \left( {t,{dt}}\right) \) from \( {\varphi \psi } \) to \( {\psi \varphi } \) is given by\n\n\[ \n\Phi {x}_{1} = {x}_{1} + {x}_{3},\;{\Phi z} = z + {x}_{1}{x}_{2} + {x}_{3}{x}_{2} + {ydt} \n\]\n\n\[ \n\Phi {w}_{1} = {w}_{1} + 2{w}_{2} - {2yzt} + 2{x}_{1}{vdt} + 2{x}_{3}{vtdt} \n\]\n\n\[ \n\Phi {w}_{2} = {w}_{2} - {yz} - {x}_{1}{x}_{2}y + {x}_{1}{x}_{2}{yt} + \left( {2{x}_{1} + {x}_{3}}\right) {vdt} \n\]\n\nand \( \Phi = \) identity on the other generators. (This is a longish, but easy computation.)\n\nOn the other hand, if \( \varphi \sim {\varphi }^{\prime } \) then \( H\left( {\varphi }^{\prime }\right) = H\left( \varphi \right) \) . This implies that \( {\varphi }^{\prime }{x}_{i} = {x}_{i} \) . Hence \( 0 = d\left( {\varphi - {\varphi }^{\prime }}\right) y \) . There are no cocycles in degree 5, so \( {\varphi }^{\prime }y = {\varphi y} = y \) . Again, because \( H\left( {\varphi }^{\prime }\right) = H\left( \varphi \right) \) we have \( {\varphi }^{\prime }z = {\varphi z} + \lambda {x}_{2}{x}_{3} \), some \( \lambda \in \mathbb{R} \) . Similarly, if \( {\psi }^{\prime } \sim \psi \) then \( {\psi }^{\prime }{x}_{i} = \psi {x}_{i},{\psi }^{\prime }y = {\psi y} \) and \( {\psi }^{\prime }z = {\psi z} + \mu {x}_{2}{x}_{3} \), some \( \mu \in \mathbb{R} \) . A computation now shows that \( {\psi }^{\prime }{\varphi }^{\prime }z \neq {\varphi }^{\prime }{\psi }^{\prime }z \) .
|
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 graded vector space and \( V = {\left\{ {V}^{i}\right\} }_{i \geq 2} \) .\n\n\[ \text{-}V = {\bigoplus }_{m = 0}^{\infty }{V}_{m}\text{and}d : {V}_{m} \rightarrow {\left( {\Lambda }^{k + 1}V\right) }_{m - 1}\text{.} \]\n\n- \( {V}_{0} = Z \) and the inclusion induces an isomorphism \( Z\overset{ \cong }{ \rightarrow }{H}^{1, * }\left( {\Lambda V}\right) \) .\n\n- \( {H}^{ \geq k + 1, * }\left( {\Lambda V}\right) = 0 \) .
|
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 \( H\left( d\right) : {V}_{m + 1}\overset{ \cong }{ \rightarrow } \) \( {H}_{m}^{k + 1, * }\left( {\Lambda {V}_{ \leq m}, d}\right) \) . Now the first three properties above are immediate, as is the fact that \( {H}^{k + 1, * }\left( {{\Lambda V}, d}\right) = 0 \) .\n\nIt remains to see that \( {H}^{ > k + 1, * }\left( {{\Lambda V}, d}\right) = 0 \) . This is proved inductively as follows. Suppose \( {v}_{1},\ldots ,{v}_{r} \in V \) satisfy \( d{v}_{1} = 0 \) and \( d{v}_{i} \in \Lambda \left( {{v}_{1},\ldots ,{v}_{i - 1}}\right) \) . Then we can divide by \( {v}_{1},\ldots ,{v}_{r} \) to obtain a quotient Sullivan algebra \( \left( {{\Lambda W},\bar{d}}\right) \) : here we may identify \( W \) with any complement of \( \mathbb{k}{v}_{1} \oplus \cdots \oplus \mathbb{k}{v}_{r} \) in \( V \) . We call \( \left( {{\Lambda W},\bar{d}}\right) \) an \( r \) -quotient, and show that:\n\n\[ \text{- For any r-quotient}\left( {{\Lambda W},\bar{d}}\right) ,{H}^{ \geq k + 1, * }\left( {{\Lambda W},\bar{d}}\right) = 0\text{.} \]\n\n(12.11)\n\nIn fact suppose (12.11) is false. Then there is a least degree \( n \) in which it fails and a least \( r = {r}_{0} \) for which it fails in that degree. Thus for any \( s \) -quotient \( \left( {{\Lambda Z},\bar{d}}\right) \) and for any \( p \geq k + 1 \) we have:\n\n\[ {H}^{p, q}\left( {{\Lambda Z},\bar{d}}\right) = 0\;\text{ if }p + q < n\text{ or if }p + q = n\text{ and }s < {r}_{0}. \]\n\n(12.12)\n\nNow suppose \( \left( {{\Lambda W},\bar{d}}\right) \) is any \( {r}_{0} \) -quotient and \( \Phi \) is a cocycle of degree \( n \) in \( {\Lambda }^{p}W \) for some \( p \geq k + 1 \) . W
|
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 \( Z = {H}^{ + } \) . Then the construction of Example 6 gives a Sullivan model of the form \( \left( {{\Lambda V}, d}\right) \) with\n\n\[ V = {H}^{ + } \oplus {V}_{1} \oplus \cdots \oplus {V}_{m} \oplus \cdots \]\n\n\[ d : {V}_{m} \rightarrow {\left( {\Lambda }^{2}V\right) }_{m - 1} \]\n\nand\n\n\[ {H}^{1, * }\left( {{\Lambda V}, d}\right) = {H}^{ + }\;\text{ and }\;{H}^{ \geq 2, * }\left( {{\Lambda V}, d}\right) = 0. \]\n\nIt follows that dividing by \( {\Lambda }^{ \geq 2}V \) and by \( {V}_{ \geq 1} \) defines a quasi-isomorphism\n\n\[ \left( {{\Lambda V}, d}\right) \overset{ \simeq }{ \rightarrow }\left( {H,0}\right) \]\n\nwhich exhibits \( \left( {{\Lambda V}, d}\right) \) as the minimal Sullivan model for \( \left( {H,0}\right) \).
|
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 right translation. Then \( {A}_{DR}{\left( G\right) }^{G} = \left( {\Lambda {\mathfrak{g}}^{\sharp }, d}\right) \), and the formula for the exterior derivative gives\n\n\[ \langle {d\omega };x, y\rangle = - \langle \omega ,\left\lbrack {x, y}\right\rbrack \rangle ,\;\begin{array}{l} \omega \in {\mathfrak{g}}^{\sharp } \\ x, y \in \mathfrak{g}. \end{array} \]\n\nBecause \( {\mathfrak{g}}^{\sharp } \) generates \( \Lambda {\mathfrak{g}}^{\sharp } \) and because \( d \) is purely quadratic this formula determines \( d \) .\n\nAlthough \( \Lambda {\mathfrak{g}}^{\sharp } \) is a free commutative graded algebra, the cochain algebra \( \left( {\Lambda {\mathfrak{g}}^{\sharp }, d}\right) \) does not, in general, satisfy the nilpotence condition required for Sullivan algebras. In fact, it is easy to see that \( \left( {\Lambda {\mathfrak{g}}^{\sharp }, d}\right) \) is a Sullivan algebra if and only if \( \mathfrak{g} \) is a nilpotent Lie algebra.\n\nOn the other hand, \( G \) is always homeomorphic to the product of a compact Lie group with some \( {\mathbb{R}}^{n} \) [32]. In particular (as with any compact manifold, [43],[69]) \( {H}_{ * }\left( {G;\mathbb{R}}\right) \) is a finite dimensional vector space. Thus the theorem of\n\nHopf (Example 3, \( §{12} \) (a)) shows that \( {H}^{ * }\left( {G;\mathbb{R}}\right) \) is an exterior algebra on a finite dimensional vector space \( P \) concentrated in odd degrees. This gives a minimal Sullivan model\n\n\[ \left( {{\Lambda P},0}\right) \overset{ \simeq }{ \rightarrow }{A}_{DR}\left( G\right) \]\n\nwhich contrasts with the morphism \( \left( {\Lambda {\mathfrak{g}}^{\sharp }, d}\right) \rightarrow {A}_{DR}\left( G\right) \) .
|
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 \( \Gamma \) -invariant differential forms on \( X \) ). Thus we may regard \( {\left( {A}_{DR}G\right) }^{G} \) as a subcochain algebra of \( {A}_{DR}X \) . In [130], K. Nomizu proved that this inclusion is a quasi-isomorphism.
|
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 \( X = {S}^{1} \times \cdots \times {S}^{1} \) . In particular, \( X \) is formal. The converse is true. Indeed, assume that there exists a quasi-isomorphism \( \left( {\Lambda \left( {{x}_{1},{x}_{2},\ldots ,{x}_{k}}\right), d}\right) \overset{\varphi }{ \rightarrow }{H}^{ * }\left( {X;\mathbb{R}}\right) \) . Then \( \varphi \) is obviously surjective, and since the product \( {x}_{1}{x}_{2}\ldots {x}_{k} \) is a cocycle which cannot be a coboundary, \( \varphi \) is injective. This in turn implies that \( d = 0 \) .
|
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, \( {A}_{DR}{\left( G/K\right) }^{G}\overset{ \simeq }{ \rightarrow }{A}_{DR}\left( {G/K}\right) \) . An argument of E. Cartan shows that the differential in \( {A}_{DR}{\left( G/K\right) }^{G} \) is zero, thereby exhibiting \( G/K \) as a formal space. It is interesting to note that this is still the only proof available of the formality of \( G/K \) . Cartan’s argument runs as follows: observe that \( \tau \) induces an involution \( \sigma \) of \( G/K \) and that \( {A}_{DR}\left( \sigma \right) \) restricts to an involution of \( {A}_{DR}{\left( G/K\right) }^{G} \) . Since the action of \( G \) on \( G/K \) is transitive, \( {A}_{DR}{\left( G/K\right) }^{G} \) may be identified with a subalgebra of \( \Lambda {T}_{\bar{e}}^{ * }\left( {G/K}\right) = \Lambda {\left( \mathfrak{g}/\mathfrak{k}\right) }^{ * } \) . Let \( {\tau }^{\prime } : {TG} \rightarrow {TG} \) and \( {\sigma }^{\prime } : {TG}/K \rightarrow {TG}/K \) denote the derivatives of \( \tau \) and \( \sigma \) . Since \( \tau \) is an involution and \( \mathfrak{k} = \left\{ {h \in \mathfrak{g} \mid {\tau }^{\prime }h = 1}\right\} \), it follows that \( {\sigma }^{\prime } = - {id} \) in \( \mathfrak{g}/\mathfrak{k} \) . Hence \( {A}_{DR}\left( \sigma \right) = {\left( -1\right) }^{p}{id} \) in \( {A}_{DR}^{p}{\left( G/K\right) }^{G} \) . Since \( {A}_{DR}\left( \sigma \right) \) commutes with \( d, d = 0 \) .
|
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 and surjection\n\n\[
{A}_{DR}\left( M\right) \overset{i}{ \leftarrow }\left( {\ker {d}^{c}, d}\right) \overset{\varrho }{ \rightarrow }\left( {\ker {d}^{c}/\operatorname{Im}{d}^{c},0}\right)
\]\n\nare quasi-isomorphisms. This exhibits \( M \) as formal.
|
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 \( \left( {\gamma ,\alpha }\right) : \left( {C{ \times }_{B}A, d}\right) \rightarrow \left( {{C}^{\prime }{ \times }_{{B}^{\prime }}{A}^{\prime }, d}\right) \) is a quasi isomorphism.
|
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\n\nto conclude that \( \left( {\gamma ,\alpha }\right) \) is an isomorphism.\n\nThe other case to consider is that \( {\varphi }^{\prime } \) and \( \psi \) are surjective. As in \( §{12}\left( \mathrm{\;b}\right) \) choose a surjective commutative cochain algebra morphism \( \sigma : \left( {E,\delta }\right) \rightarrow \left( {{B}^{\prime }, d}\right) \) with \( H\left( E\right) = \mathbb{k} \), and consider\n\n\n\nThe argument above shows that \( \left( {\gamma ,{\lambda \alpha }}\right) : \left( {C{ \times }_{B}A, d}\right) \rightarrow \left( {{C}^{\prime }{ \times }_{{B}^{\prime }}\left( {{A}^{\prime } \otimes E}\right), d}\right) \) is a quasi-isomorphism. Since \( {\varphi }^{\prime } \) is surjective the same argument shows that \( \left( {{id},\lambda }\right) : \left( {{C}^{\prime }{ \times }_{{B}^{\prime }}{A}^{\prime }, d}\right) \rightarrow \left( {{C}^{\prime }{ \times }_{{B}^{\prime }}\left( {{A}^{\prime } \otimes E}\right), d}\right) \) is a quasi-isomorphism. Hence so is \( \left( {\gamma ,\alpha }\right) \) .
|
Yes
|
Lemma 13.4 Suppose \( \alpha ,\beta ,\gamma \) 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 \( {\gamma }^{\prime } \) makes the diagram above commutative. Thus in this case the Lemma follows from Lemma 13.3.\n\nTo construct \( {\alpha }^{\prime } \), let \( \Psi : \left( {{\Lambda W}, d}\right) \rightarrow \left( {G, d}\right) \otimes \Lambda \left( {t,{dt}}\right) \) be a homotopy from \( {\beta \psi } \) to \( {\eta \alpha } : {\varepsilon }_{0}\Phi = {\beta \psi } \) and \( {\varepsilon }_{1}\Psi = {\eta \alpha } \), as described in \( §{12}\left( \mathrm{\;b}\right) \) . Use \( {\varepsilon }_{1} \) to form the fibre product \( G \otimes \Lambda \left( {t,{dt}}\right) { \times }_{G}E \) . Then, in the diagram of cochain algebra morphisms\n\n\[ \left( {{\Lambda W}, d}\right) \xrightarrow[\left( \Phi ,\alpha \right) ]{}G \otimes \Lambda \left( {t,{dt}}\right) { \times }_{G}F, \]\n\n\nthe vertical arrow is a surjective quasi-isomorphism. Use the Lifting lemma 12.4 to lift \( \left( {\Phi ,\alpha }\right) \) to \( \Psi : \left( {{\Lambda W}, d}\right) \rightarrow \bar{F} \otimes \Lambda \left( {t,{dt}}\right) \), and set \( {\alpha }^{\prime } = {\varepsilon }_{0}\Psi \) . The morphism \( {\gamma }^{\prime } \) is constructed in the same way.\n\nFinally suppose \( \xi \) (for definiteness) is not surjective. Use the surjective trick \( \left( {§{12}\left( \mathrm{\;b}\right) }\right) \) to extend \( \xi \) to a surjection \( \xi \cdot \sigma : \left( {C, d}\right) \otimes \left( {E,\delta }\right) \rightarrow \left( {G, d}\right) \), with \( H\left( {E,\delta }\right) = \mathbb{R} \) . Then the fibre square for \( \xi \cdot \sigma \) and \( \eta \) is weakly equivalent to the fibre square for \( \xi ,\eta \) by Lemma 13.3 and weakly equivalent to the fibre square for \( \varphi ,\psi \) by the argument above.
|
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 model for the adjunction space.
|
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\n\nthe central arrow is a quasi-isomorphism.
|
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 a commutative model for \( X{ \cup }_{f}{CY} \) .
|
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 of \( \{ 1\} \) in \( I \), and notice that the chain of quasi-isomorphisms\n\n\[ \n{A}_{PL}\left( {{CY},{pt}}\right) \overset{ \simeq }{ \rightarrow }{A}_{PL}\left( {Y \times I, Y\times \{ 0\} }\right) \overset{ \simeq }{ \leftarrow }{A}_{PL}\left( Y\right) \otimes {A}_{PL}\left( {I,\{ 0\} }\right) \n\]\n\nis compatible with the surjections \( {A}_{PL}\left( {i}_{1}\right) \) and \( {id} \otimes {A}_{PL}\left( {i}_{1}\right) \) . Regard the identity of \( I \) as a 1-simplex. Restriction to this simplex is a quasi-isomorphism \( {A}_{PL}\left( {I,\{ 0\} }\right) \rightarrow {\Lambda }^{ + }\left( {t,{dt}}\right) \) which converts \( {id} \otimes {A}_{PL}\left( {i}_{1}\right) \) to \( {id} \otimes {\varepsilon }_{1} \) . Since \( {A}_{PL}\left( {CY}\right) \) \( = \mathbb{k} \oplus {A}_{PL}\left( {{CY},{pt}}\right) \) we obtain a chain of quasi-isomorphisms connecting \( {A}_{PL}\left( X\right) { \times }_{{A}_{PL}\left( Y\right) }{A}_{PL}\left( {CY}\right) \) with \( {A}_{PL}\left( X\right) { \times }_{{A}_{PL}\left( Y\right) }\left( {\mathbb{k} \oplus \left\lbrack {{A}_{PL}\left( Y\right) \otimes {\Lambda }^{ + }\left( {t,{dt}}\right) }\right\rbrack }\right) . \n\nOn the other hand, there are homotopics \( {m}_{\alpha }{\varphi }_{\alpha } \sim {A}_{PL}\left( {f}_{\alpha }\right) {m}_{X} \) . A choice of\nthese defines a commutative diagram\n\n\n\nand this connects \( {A}_{PL}\left( X\right) { \times }_{{A}_{PL}\left( Y\right) }\left( {\mathbb{k} \oplus \left\lbrack {{A}_{PL}\left( Y\right) \otimes {\Lambda }^{ + }\left( {t,{dt}}\right) }\right\rbrack }\right) \) to \( {\Lambda V}{ \times }_{B}\left( {\mathbb{R} \oplus \left\lbrack {B \otimes {\Lambda }^{ + }\left( {t,{dt}}\right) }\right\rbrack }\right) \) by a chain of quasi-isomorphisms too.
|
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 \( \mathbb{K}{ \times }_{B}\left( {\mathbb{R} \oplus \left\lbrack {B \otimes {\Lambda }^{ + }\left( {t,{dt}}\right) }\right\rbrack }\right) \) . But this is just the cochain algebra \( \mathbb{K} \oplus \left\lbrack {B \otimes J}\right\rbrack \), where \( J = \ker {\varepsilon }_{1} \cap {\Lambda }^{ + }\left( {t,{dt}}\right) \) . Now \( J \) is the ideal generated by \( t\left( {1 - t}\right) \) and \( {dt} \) and thus the inclusion \( \mathbb{R}{dt} \rightarrow J \) is a quasi-isomorphism. It follows that \( \mathbb{R} \oplus \left\lbrack {B \otimes {dt}}\right\rbrack \) includes quasi-isomorphically in \( \mathbb{R} \oplus \left\lbrack {B \otimes J}\right\rbrack \) . Let \( H \subset B \otimes {dt} \) be any subspace of cocycles mapping isomorphically to \( H\left( {B \otimes {dt}}\right) \) . Since \( \left( {B \otimes {dt}}\right) \cdot \left( {B \otimes {dt}}\right) \subset B \otimes \left( {{dt} \land {dt}}\right) = 0 \), the inclusion \( \left( {\mathbb{R} \oplus H,0}\right) \rightarrow \mathbb{R} \oplus \left\lbrack {B \otimes J}\right\rbrack \) is a cochain algebra quasi-isomorphism, exhibiting \( \mathbb{R} \oplus H \) as a commutative model for \( \{ {pt}\} { \cup }_{f}{CY} \) in which \( H \cdot H = 0 \) . Finally, \( \{ {pt}\} { \cup }_{f}{CY} = {CY}/\left( {Y\times \{ 1\} }\right) \), which is homotopy equivalent to \( {\sum Y} \) since \( \left( {Y,{y}_{0}}\right) \) is well-based \( \left( {§1\left( \mathrm{\;d}\right) }\right) \) .
|
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 \( {\Lambda V}{ \times }_{\Lambda W}\left( {\mathbb{k} \oplus \left\lbrack {{\Lambda }^{ + }W \otimes {\Lambda }^{ + }\left( {t,{dt}}\right) }\right\rbrack }\right) \) , where \( \left( {{\Lambda W}, d}\right) \) is the minimal Sullivan model for \( {S}^{n} \) (Example 1, \( §{12}\left( \mathrm{a}\right) \) ). The quasi-isomorphism \( \left( {{\Lambda W}, d}\right) \rightarrow \left( {H\left( {S}^{n}\right) ,0}\right) \) then defines a quasi-isomorphism \( {\Lambda V}{ \times }_{\Lambda W}\left( {\mathbb{I}k \oplus \left\lbrack {{\Lambda }^{ + }W \otimes {\Lambda }^{ + }\left( {t,{dt}}\right) }\right\rbrack }\right) \overset{ \simeq }{ \rightarrow }{\Lambda V}{ \times }_{H\left( {S}^{n}\right) }\left( {\mathbb{I}k \oplus \left\lbrack {{H}^{ + }\left( {S}^{n}\right) \otimes {\Lambda }^{ + }\left( {t,{dt}}\right) }\right\rbrack }\right) . \n\nThese constructions are made using a Sullivan representative \( {\varphi }_{a} : \left( {{\Lambda V}, d}\right) \rightarrow \) \( \left( {{\Lambda W}, d}\right) \) for \( a \) . Let \( \varphi \) be the composite of \( {\varphi }_{a} \) with \( \bar{m} : \left( {{\Lambda W}, d}\right) \overset{ \simeq }{ \rightarrow }H\left( {S}^{n}\right) \) and define a linear inclusion \( \lambda \) of \( {\Lambda V} \) into the fibre product by\n\n\[ \lambda \left( 1\right) = 1\text{ and }{\lambda \Phi } = \left( {\Phi ,{\varphi \Phi } \otimes t}\right) ,\;\Phi \in {\Lambda }^{ + }V. \]\n\nNow \( {H}^{ + }\left( {S}^{n}\right) = \mathbb{R}\left\lbrack e\right\rbrack \), where \( \left\lbrack e\right\rbrack \) is dual to the fundamental class \( \left\lbrack {S}^{n}\right\rbrack \) . Hence \( {\lambda \Phi } = \left( {\Phi ,0}\right) \) if \( \Phi \in {\Lambda }^{ \geq 2}V \) or if \( \Phi \in {V}^{k}, k \neq n \) . Moreover, for \( v \in {V}^{n} \) ,\n\n\[ {\lambda v} = \left( {v,\langle v;\alpha \rangle \left\lbrack e\right\rbrack \otimes t}\right) . \]\n\nIt follows at once that \( \lambda \) is an algebra morphism and that it can be extended to a cochain algebra morphism\n\n\[ \left( {{\Lambda V} \oplus \mathbb{R}u,{d}_{\alpha }}\right) \rightarrow {\Lambda V}{ \times }_{H\left( {S}^{n}\right) }\left( {\mathbb{R} \oplus \left\lbrack {{H}^{ + }\left( {S}^{n}\right) \otimes {\Lambda }^{ + }\left( {t,{dt}}\right) }\right\rbrack }\right) \]\n\nby defining \( u \mapsto {\left( -1\right) }^{n}\left\lbrack e\right\rbrack \otimes {dt} \) . It is a trivial verification that this is a quasi-isomorphism.
|
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\[ \left\langle {{e}^{\prime };{\left\lbrack \alpha ,\alpha \right\rbrack }_{W}}\right\rangle = - \left\langle {{e}^{2};\alpha ,\alpha }\right\rangle = - 2. \]\n\nIn particular, \( {\left\lbrack \alpha ,\alpha \right\rbrack }_{W} \) is not a torsion class in \( {\pi }_{{4n} - 1}\left( {S}^{2n}\right) \).
|
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 \( {S}^{3} \vee {S}^{3} \) by the maps \( {\left\lbrack {a}_{0},{\left\lbrack {a}_{0},{a}_{1}\right\rbrack }_{W}\right\rbrack }_{W} \) and \( {\left\lbrack {a}_{1},{\left\lbrack {a}_{1},{a}_{0}\right\rbrack }_{W}\right\rbrack }_{W} \) . The same argument as in Proposition 13.12 shows that the resulting space has a commutative model of the form \( \left( {{\Lambda V} \oplus \mathbf{k}{u}_{0} \oplus \mathbf{k}{u}_{1}, D}\right) \) where \( {u}_{0}{u}_{1} = {u}_{0}^{2} = {u}_{1}^{2} = {u}_{0}{\Lambda }^{ + }V = {u}_{1}{\Lambda }^{ + }V = 0 \) , \( D{u}_{0} = D{u}_{1} = 0 \), and \( {Dv} = {dv} + \left\langle {v;{\left\lbrack {\alpha }_{0},{\left\lbrack {\alpha }_{0},{\alpha }_{1}\right\rbrack }_{W}\right\rbrack }_{W}}\right\rangle {u}_{0} + \left\langle {v;{\left\lbrack {\alpha }_{1},{\left\lbrack {\alpha }_{1},{\alpha }_{0}\right\rbrack }_{W}\right\rbrack }_{W}}\right\rangle {u}_{1}. \) Of course \( \deg {u}_{0} = 8 = \deg {u}_{1} \) . Now \( \left( {{\Lambda V}, d}\right) = \left( {\Lambda \left( {{e}_{0},{e}_{1}, x,{y}_{0},{y}_{1},{z}_{1},{z}_{2},\ldots }\right), d}\right) \) with \[ {dx} = {e}_{0}{e}_{1}\;d{y}_{0} = {e}_{0}x, d{y}_{1} = {e}_{1}x \] and \( \deg {z}_{i} \geq 9, i = 1,2,\ldots \) . From Proposition 13.16 it follows that \( D{y}_{0} = \) \( {e}_{0}x + {u}_{0}, D{y}_{1} = {e}_{1}x + {u}_{1} \) and \( D = d \) on the other basis elements of \( V \) . Moreover, it follows from diagram 13.15 that \( {u}_{0} \) and \( {u}_{1} \) represent the cohomology classes dual to the cells \( {D}_{0}^{8} \) and \( {D}_{1}^{8} \), so that \( \left\lbrack 1\right\rbrack ,\left\lbrack {e}_{0}\right\rbrack ,\left\lbrack {e}_{1}\right\rbrack ,\left\lbrack {u}_{0}\right\rbrack \) and \( \left\lbrack {u}_{1}\right\rbrack \) is a basis for \( H\left( {{\Lambda V} \oplus \mathbb{k}{u}_{0} \oplus \mathbb{k}{u}_{1}, D}\right) \) . This implies that a quasi-isomorphism \[ \left( {{\Lambda V} \oplus \mathbb{k}{u}_{0} + \mathbb{k}{u}_{1}, D}\right) \overset{ \simeq }{ \rightarrow }\left( {\Lambda \left( {{e}_{0},{e}_{1}, x}\right) /{e}_{0}{e}_{1}x, d}\right) \] is given by \( {e}_{0} \mapsto {e}_{0},{e}_{1} \mapsto {e}_{1}, x \mapsto x,{y}_{i} \mapsto 0,{z}_{i} \mapsto 0,{u}_{0} \mapsto - {e}_{0}x \), and \( {u}_{1} \mapsto - {e}_{1}x \) . Thus \( \left( {\Lambda \left( {{e}_{0},{e}_{1}, x}\right) /{e}_{0}{e}_{1}x, d}\right) \) is a commutative model for \( {S}^{3} \vee {S}^{3} \cup \left( {{D}_{0}^{8} \coprod {D}_{1}^{8}}\right) \) .
|
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 \( B\left( k\right) = B\left( {k - 1}\right) \otimes \Lambda {V}_{k} \) and \( d : {V}_{k} \rightarrow B\left( {k - 1}\right) \) . Hence\n\n\[ \frac{\left( B\left( k - 1\right) \otimes {\Lambda }^{ \leq n}{V}_{k}, d\right) }{\left( B\left( k - 1\right) \otimes {\Lambda }^{ < n}{V}_{k}, d\right) } = \left( {B\left( {k - 1}\right), d}\right) \otimes \left( {{\Lambda }^{n}{V}_{k},0}\right) . \]\n\nAssume by induction that \( \left( {B\left( {k - 1}\right), d}\right) \) is \( \left( {B, d}\right) \) -semifree. Then this equation identifies the quotient on the left as \( \left( {B, d}\right) \) -semifree for each \( n \geq 1 \) . It follows from Lemma 6.3 that each \( \left( {B\left( k\right), d}\right) \) and \( \left( {B\left( k\right), d}\right) /\left( {B\left( {k - 1}\right), d}\right) \) are \( \left( {B, d}\right) - \) semifree. Now a second application of Lemma 6.3 shows that \( \left( {B \otimes {\Lambda V}, d}\right) = \) \( \mathop{\bigcup }\limits_{k}\left( {B\left( k\right), d}\right) \) is \( \left( {B, d}\right) \) -semifree too.
|
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 algebra and the inclusion \( \varphi : \left( {{B}_{1}, d}\right) \rightarrow \left( {B, d}\right) \) is a quasi-isomorphism. In particular the restriction \( {\varphi }_{1} : \left( {{B}_{1}, d}\right) \rightarrow \left( {C, d}\right) \) of \( \varphi \) satisfies: \( {H}^{1}\left( {\varphi }_{1}\right) \) is injective.\n\nNow because \( {\left( {B}_{1}\right) }^{0} = \mathbb{R} \) the argument of Proposition 12.1 applies verbatim to show the existence of a Sullivan model \( {m}_{1} : \left( {{B}_{1} \otimes {\Lambda V}, d}\right) \overset{ \simeq }{ \rightarrow }\left( {C, d}\right) \) for \( {\varphi }_{1} \) . Thus a commutative diagram of cochain algebra morphisms is given by\n\n\n\nin which \( j\left( z\right) = 1{ \otimes }_{{B}_{1}}z \) .\n\nThis may be rewritten as\n\n\n\nLemma 14.2 and the preceding remarks show that \( i \otimes {id} \) is a quasi-isomorphism and \( \left( {B \otimes {\Lambda V}, d}\right) \) is a Sullivan algebra; hence \( m : \left( {B \otimes {\Lambda V}, d}\right) \overset{ \simeq }{ \rightarrow }\left( {C, d}\right) \) is a Sullivan model for \( \varphi \) .
|
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\[ \Phi \left( z\right) = {\varphi }_{0}\left( z\right) + \left( {{\varphi }_{1}\left( z\right) - {\varphi }_{0}\left( z\right) }\right) t + {\left( -1\right) }^{\deg z}h\left( z\right) {dt} + \Omega ,\]\n\nwhere \( \Omega \in A \otimes \left( {I + {dI}}\right), I \subset \Lambda \left( t\right) \) denoting the ideal generated by \( t\left( {1 - t}\right) \) . Then, because \( {d\Phi } = {\Phi d} \), we obtain \( {\varphi }_{0} - {\varphi }_{1} = {dh} + {hd} \) . Moreover, since \( \Phi \) restricts to \( \alpha \) in \( B \) it follows that \( \Phi \left( {bz}\right) = \alpha \left( b\right) \Phi \left( z\right), b \in B \) . Hence \( h\left( {bz}\right) = {\left( -1\right) }^{\deg b}b \cdot h\left( z\right) \) ; i.e., \( h \) is \( B \) -linear.
|
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 \( \alpha \) to the morphism \( \tau = i{d}_{B} \cdot \alpha : B \otimes {\Lambda V} \rightarrow \) \( B \otimes {\Lambda W} \) . Then \( {\sigma \tau } = {id} \) .\n\nNow suppose by induction that \( \sigma \) and \( \tau \) restrict to inverse isomorphisms between \( B \otimes \Lambda {W}^{ < n} \) and \( B \otimes \Lambda {V}^{ < n} \) . Let \( w \in {W}^{n} \) . Then \( {\tau w} = {w}^{\prime } + u \) with \( {w}^{\prime } \in {W}^{n} \) and \( u \in {B}^{ + } \otimes \Lambda {W}^{ < n} \) . Hence \( w = {\sigma \tau w} = \sigma {w}^{\prime } + {\sigma u} = {w}^{\prime } + {\sigma u} \) . In other words,\n\n\[ w - {w}^{\prime } = {\sigma u} \in W \cap \left( {{B}^{ + } \otimes \Lambda {V}^{ < n}}\right) = 0. \]\n\nBy induction, \( \sigma \) is injective in \( B \otimes \Lambda {V}^{ < n} \) . Hence \( w = {w}^{\prime } \) and \( u = 0 \) ; i.e., \( {\tau w} = w \) .\n\nThis shows that \( {\tau \sigma } = {id} \) in \( {W}^{n} \) ; which closes the induction.
|
Yes
|
Lemma 14.8 There is a relative Sullivan algebra \( \left( {{B}_{1} \otimes {\Lambda V},{d}^{\prime }}\right) \) and an isomorphism, \[ \sigma : \left( {B \otimes {\Lambda V},{d}^{\prime }}\right) = \left( {B, d}\right) { \otimes }_{\left( {B}_{1}, d\right) }\left( {{B}_{1} \otimes {\Lambda V},{d}^{\prime }}\right) \overset{ \cong }{ \rightarrow }\left( {B \otimes {\Lambda V}, d}\right) , \] restricting to the identity in \( B \) .
|
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 induction we have constructed \( \left( {{B}_{1} \otimes {\Lambda V}\left( {k - 1}\right) ,{d}^{\prime }}\right) \) and \[ \sigma : \left( {B \otimes {\Lambda V}\left( {k - 1}\right) ,{d}^{\prime }}\right) \overset{ \cong }{ \rightarrow }\left( {B \otimes {\Lambda V}\left( {k - 1}\right), d}\right) . \] Lemma 14.2, applied to the inclusion \( \left( {{B}_{1}, d}\right) \rightarrow \left( {B, d}\right) \) shows that the inclusion \( \left( {{B}_{1} \otimes {\Lambda V}\left( {k - 1}\right) ,{d}^{\prime }}\right) \rightarrow \left( {B \otimes {\Lambda V}\left( {k - 1}\right) ,{d}^{\prime }}\right) \) is a quasi-isomorphism. Hence if \( \left\{ {v}_{\alpha }\right\} \) is a basis of \( {V}_{k} \) there are \( {d}^{\prime } \) -cocycles \( {z}_{\alpha } \in {B}_{1} \otimes {\Lambda V}\left( {k - 1}\right) \) and elements \( {y}_{\alpha } \in B \otimes {\Lambda V}\left( {k - 1}\right) \) such that \( d{v}_{\alpha } = \sigma \left( {{z}_{\alpha } + {d}^{\prime }{y}_{\alpha }}\right) \) . Extend \( {d}^{\prime } \) and \( \sigma \) to \( {V}_{k} \) by setting \( {d}^{\prime }{v}_{\alpha } = {z}_{\alpha } \) and \( \sigma {v}_{\alpha } = {v}_{\alpha } - \sigma {y}_{\alpha } \) . By construction, \( \sigma \) is a cochain algebra morphism. By induction, it restricts to an isomorphism of \( B \otimes {\Lambda V}\left( {k - 1}\right) \) onto itself. Denote the inverse of the restriction by \( \tau \), extend \( \tau \) to \( B \otimes {\Lambda V}\left( k\right) \) by setting \( \tau {v}_{\alpha } = {v}_{\alpha } + {y}_{\alpha } \), and verify that \( \tau \) and \( \sigma \) are inverse isomorphisms of \( B \otimes {\Lambda V}\left( k\right) \) .
|
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 \( \gamma \) : \( \left( {B \otimes {\Lambda V}, d}\right) \rightarrow \left( {{B}^{\prime } \otimes \Lambda {V}^{\prime }, d}\right) \) such that \( {\eta \gamma } = {id} \) .\n\nBy the very definition of minimality, \( d : V \rightarrow {B}^{ + } \otimes {\Lambda V} \oplus {\Lambda }^{ \geq 2}V \) . Thus if \( V = \mathop{\bigcup }\limits_{k}V\left( k\right) \) with \( V\left( k\right) = V\left( {k - 1}\right) \oplus {V}_{k} \) and \( d : {V}_{k} \rightarrow B \otimes {\Lambda V}\left( {k - 1}\right) \) it follows that \( d : {V}_{k}^{n} \rightarrow B \otimes \Lambda {V}^{ < n} \otimes \Lambda \left( {V{\left( k - 1\right) }^{n}}\right) \) . Thus to construct \( \gamma \) it is enough to assume it has been defined in \( A = B \otimes \Lambda {V}^{ < n} \otimes \Lambda \left( {V{\left( k - 1\right) }^{n}}\right) \) and to extend it to \( A \otimes \Lambda {V}_{k}^{n} \) .\n\nObserve that \( \eta \) factors to give a quasi-isomorphism of cochain complexes,\n\n\[ \frac{{B}^{\prime } \otimes \Lambda {V}^{\prime }}{\gamma \left( A\right) }\underset{ \simeq }{\overset{\bar{\eta }}{ \rightarrow }}\frac{B \otimes {\Lambda V}}{A} \]\n\n(we are dividing by graded subspaces, not ideals). The cochain complex \( B \otimes \) \( {\Lambda V}/A \) contains no elements of degree \( n - 1 \), and hence no coboundaries of degree \( n \) . Thus every cocycle of degree \( n \) is the image, under \( \bar{\eta } \), of a cocycle of degree \( n \) in \( \left( {{B}^{\prime } \otimes \Lambda {V}^{\prime }}\right) /\gamma \left( A\right) \) . In particular, if \( \left\{ {v}_{\alpha }\right\} \) is a basis of \( {V}_{k}^{n} \) then there are elements \( {x}_{\alpha } \in {B}^{\prime } \otimes \Lambda {V}^{\prime } \) and elements \( {a}_{\alpha },{a}_{\alpha }^{\prime } \in A \) such that\n\n\[ \eta {x}_{\alpha } = {v}_{\alpha } + {a}_{\alpha }\;\text{ and }\;d{x}_{\alpha } = \gamma \left( {a}_{\alpha }^{\prime }\right) . \]\n\nHence \( d{v}_{\alpha } = {d\eta }\left( {{x}_{\alpha } - \gamma {a}_{\alpha }}\right) = {a}_{\alpha }^{\prime } - d{a}_{\alpha } \) . Extend \( \gamma \) to \( {V}_{k}^{n} \) by putting \( \gamma {v}_{\alpha } = \) \( {x}_{\alpha } - \gamma {a}_{\alpha }. \)\n\nThis completes the construction of \( \gamma \) . Since \( {\eta \gamma } = {id},\gamma \) is also a quasi-isomorphism. Now the same argument applied to \( \gamma \) gives a morphism \( \chi \) : \( \left( {{B}^{\prime } \otimes \Lambda {V}^{\prime }, d}\right) \rightarrow \left( {B \otimes {\Lambda V}, d}\right) \) such that \( {\gamma \chi } = {id} \) . Thus \( \gamma \) is both injective and surjective; i.e., it is an isomorphism. Since \( {\eta \gamma } = {id},\eta \) is the isomorphism \( {\gamma }^{-1} \) .
|
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 second minimal Sullivan model for \( \varphi \) then there is an isomorphism\n\n\[ \alpha : \left( {B \otimes {\Lambda V}, d}\right) \overset{ \cong }{ \rightarrow }\left( {B \otimes \Lambda {V}^{\prime }, d}\right) \]\n\nrestricting to \( i{d}_{B} \), and such that \( {m}^{\prime }\alpha \sim m \) rel \( B \) .
|
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\n\n\n\nto extend \( i{d}_{B} \) to a morphism \( \alpha : \left( {B \otimes {\Lambda V}, d}\right) \rightarrow \left( {B \otimes \Lambda {V}^{\prime }, d}\right) \) such that \( {m}^{\prime }\alpha \sim \) \( m \) rel \( B \) . Now Theorem 14.11 asserts that \( \alpha \) is an isomorphism.
|
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 \( \left( {\Lambda \bar{W}, d}\right) \) minimal Sullivan algebras and \( \left( {E, d}\right) \) and \( \left( {F, d}\right) \) contractible Sullivan algebras. Let \( \psi \) be the composite \( \left( {\Lambda \bar{V}, d}\right) \rightarrow \left( {{\Lambda V}, d}\right) \overset{\varphi }{ \rightarrow }\left( {{\Lambda W}, d}\right) \rightarrow \) \( \left( {\Lambda \bar{W}, d}\right) \) . Then we may identify \( H\left( \varphi \right) = H\left( \psi \right) \) and \( H\left( {Q\left( \varphi \right) }\right) = Q\left( \psi \right) \) . But by Theorem 14.11, \( \psi \) is a quasi-isomorphism if and only if it is an isomorphism. If \( \psi \) is an isomorphism so is \( Q\left( \psi \right) \) . Conversely, suppose \( Q\left( \psi \right) \) is an isomorphism, then \( \psi \) induces isomorphisms \( {\Lambda }^{ \geq m}\bar{V}/{\Lambda }^{ > m}\bar{V}\overset{ \simeq }{ \rightarrow }{\Lambda }^{ \geq m}\bar{W}/{\Lambda }^{ > m}\bar{W} \) . By induction it induces isomorphisms \( \Lambda \bar{V}/{\Lambda }^{ > m}\bar{V}\overset{ \cong }{ \rightarrow }\Lambda \bar{W}/{\Lambda }^{ > m}\bar{W} \) . Since \( {\left( \Lambda \bar{V}\right) }^{k} = {\left( \Lambda \bar{V}/{\Lambda }^{ > m}\bar{V}\right) }^{k} \) for \( m > k \) it follows that \( \psi \) is an isomorphism.
|
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 \( m : \left( {B \otimes {\Lambda W}, d}\right) \overset{ \simeq }{ \rightarrow }\left( {A, d}\right) \) and recall that \( \left( {{\Lambda W},\bar{d}}\right) = \mathbb{k}{ \otimes }_{B}\left( {B \otimes {\Lambda W}, d}\right) \) is the Sullivan fibre of \( \varphi \) at an augmentation \( \varepsilon : B \rightarrow \mathbb{R} \) .\n\nWe construct a commutative model for \( \left( {{\Lambda W},\bar{d}}\right) \) as follows. Let \( \left( {B \otimes {\Lambda V}, d}\right) \) be an acyclic closure for \( \left( {B, d}\right) \) . Since \( - { \otimes }_{B}\left( {M, d}\right) \) preserves quasi-isomorphisms for any \( \left( {B, d}\right) \) -semifree module \( \left( {M, d}\right) \) - cf. Proposition \( {6.7} \) - it follows that\n\n\[ \left( {A \otimes {\Lambda V}, d}\right) \underset{ \simeq }{\overset{{id}{ \otimes }_{B}m}{ \leftarrow }}\left( {B \otimes {\Lambda V}}\right) { \otimes }_{B}\left( {B \otimes {\Lambda W}}\right) \underset{ \simeq }{\overset{\varepsilon { \otimes }_{B}{id}}{ \rightarrow }}\left( {{\Lambda W},\bar{d}}\right) \]\n\nare quasi-isomorphisms; i.e.\n\n\[ \left( {A \otimes {\Lambda V}, d}\right) = A{ \otimes }_{B}\left( {B \otimes {\Lambda V}, d}\right) \]\n\nis a commutative model for the Sullivan fibre \( \left( {{\Lambda W},\bar{d}}\right) \) .
|
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 corresponding diagram with \( {C}^{ * }\left( -\right) \) replacing \( {A}_{PL}\left( -\right) \) . But the first assertion is just Lemma 14.1 and the second follows from the natural cochain algebra quasi-isomorphisms \( {C}^{ * }\left( -\right) \overset{ \cong }{ \rightarrow } \bullet \overset{ \cong }{ \leftarrow } \) \( {A}_{PL}\left( -\right) \) of Corollary 10.10. Thus we may apply Theorem 7.10 and it asserts precisely that \( \bar{m} \) is a quasi-isomorphism.\n\nNow suppose only that \( p \) is a Serre fibration. In the diagram constructed in \( §2\left( \mathrm{c}\right) \) ,\n\n\n\n\[ ,{\lambda x} = \left( {x,\text{ const. path at }{px}}\right) ,\]\n\n\( \lambda \) is a map from a Serre fibration to a fibration. Since \( \lambda \) is a homotopy equivalence it restricts to a weak homotopy equivalence \( \bar{\lambda } : F\overset{ \simeq }{ \rightarrow }X{ \times }_{Y}{PY} \) (Proposition \( {2.5} \) (ii)).\n\nMoreover, \( {A}_{PL}\left( \lambda \right) \) is a surjective quasi-isomorphism. Apply the Lifting lemma 14.4 to the diagram\n\n\n\nto construct the quasi-isomorphism \( n \) extending \( {A}_{PL}\left( q\right) \) . Then \( \bar{n} : \left( {{\Lambda V},\bar{d}}\right) \rightarrow \) \( {A}_{PL}\left( {X{ \times }_{Y}{PY}}\right) \) is a quasi-isomorphism by the argument for fibrations, and \( \bar{m} \) is the quasi-isomorphism \( {A}_{PL}\left( \bar{\lambda }\right) \bar{n} \) .
|
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 an increasing sequence of subspaces \( 0 = W\left( {-1}\right) \subset W\left( 0\right) \subset \) \( \cdots W \) by setting \( W\left( {\ell + 1}\right) = \{ w \in W \mid {dw} \in {\Lambda W}\left( \ell \right) \} \) . It suffices to show that \( W = \mathop{\bigcup }\limits_{\ell }W\left( \ell \right) \n\nNow \( {V}_{Y} = \mathop{\bigcup }\limits_{k}{V}_{Y}\left( k\right) \) and \( V = \mathop{\bigcup }\limits_{k}V\left( k\right) \) with \( d : {V}_{Y}\left( k\right) \rightarrow \Lambda {V}_{Y}\left( {k - 1}\right) \) and \( d : V\left( k\right) \rightarrow \Lambda {V}_{Y} \otimes {\Lambda V}\left( {k - 1}\right) \) . Thus \( {V}_{Y}\left( k\right) \subset W\left( k\right) \) . If \( V\left( {k - 1}\right) \subset \mathop{\bigcup }\limits_{\ell }W\left( \ell \right) \) and \( v \in V\left( k\right) \) then since \( {dv} \in \Lambda {V}_{Y} \otimes {\Lambda V}\left( {k - 1}\right) ,{dv} \) is in some \( {\Lambda W}\left( \ell \right) \) . Hence \( v \) is in some \( W\left( {\ell + 1}\right) \) and \( V\left( k\right) \subset \bigcup W\left( \ell \right) \) as well.\n\n(iii) The morphism \( \bar{m} \) . Write\n\n\[ \n{A}_{PL}\left( Y\right) { \otimes }_{\left( \Lambda {V}_{Y}, d\right) }\left( {\Lambda {V}_{Y} \otimes {\Lambda V}, d}\right) = \left( {{A}_{PL}\left( Y\right) \otimes {\Lambda V}, d}\right) .\n\]\n\nThis is a relative Sullivan algebra, and \( m \) factors as\n\n\[ \n\left( {\Lambda {V}_{Y} \otimes {\Lambda V}, d}\right) \xrightarrow[]{{m}_{Y} \otimes {id}}\left( {{A}_{PL}\left( Y\right) \otimes {\Lambda V}, d}\right) \xrightarrow[]{{A}_{PL}\left( p\right) \cdot m}{A}_{PL}\left( X\right) .\n\]\n\nBut \( {m}_{Y}. \otimes {id} \) is a quasi-isomorphism (Lemma 14.2). Hence so is \( {A}_{PL}\left( p\right) \cdot m \) . Now apply Theorem 15.3.\n\nSince the morphism \( {A}_{PL}\left( p\right) {m}_{Y} \) has a minimal Sullivan model (Theorem 14.12) the relative Sullivan algebra \( \left( {\Lambda {V}_{Y}, d}\right) \rightarrow \left( {\Lambda {V}_{Y} \otimes {\Lambda V}, d}\right) \) may be taken to be minimal. In this case \( \left( {{\Lambda V},\bar{d}}\right) \) is minimal and \( \bar{m} : \left( {{\Lambda V},\bar{d}}\right) \overset{ \simeq }{ \rightarrow }{A}_{PL}\left( F\right) \) is the minimal model of \( F \) . Thus Proposition 15.5 has the:
|
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 \( m : \left( {\Lambda {V}_{Y} \otimes {\Lambda V}, d}\right) \overset{ \simeq }{ \rightarrow }{A}_{PL}\left( X\right) \) be the Sullivan model of diagram (15.4). Use Proposition 14.6 to construct a morphism of cochain algebras \( \varphi : \left( {\Lambda {V}_{Y} \otimes }\right. \) \( {\Lambda W}, d) \rightarrow \left( {\Lambda {V}_{Y} \otimes {\Lambda V}, d}\right) \) such that\n\n\[ \varphi = {id}\text{ in }\Lambda {V}_{Y}\;\text{ and }\;{m\varphi } \sim n\text{ rel }\left( {\Lambda {V}_{Y}, d}\right) .\n\]\n\nNow apply \( \mathbb{k}{ \otimes }_{\Lambda {V}_{Y}} \) - to obtain a morphism \( \bar{\varphi } : \left( {{\Lambda W},\bar{d}}\right) \rightarrow \left( {{\Lambda V},\bar{d}}\right) \) such that \( \bar{m}\bar{\varphi } \sim \bar{n} \) . Thus \( H\left( \bar{m}\right) H\left( \bar{\varphi }\right) = H\left( \bar{n}\right) \) . Since \( H\left( \bar{m}\right) \) is an isomorphism (Proposition 15.5) so is \( H\left( \bar{\varphi }\right) \) . Put \( I\left( k\right) = \left( {{\Lambda }^{ \geq k}{V}_{Y} \otimes {\Lambda W}, d}\right) \) and \( J\left( k\right) = \) \( \left( {{\Lambda }^{ \geq k}{V}_{Y} \otimes {\Lambda V}, d}\right) \) . Then \( \varphi \) restricts to maps \( I\left( k\right) \rightarrow J\left( k\right) \) . The induced maps \( I\left( k\right) /I\left( {k + 1}\right) \rightarrow J\left( k\right) /J\left( {k + 1}\right) \) have the form\n\n\[ {id} \otimes \bar{\varphi } : \left( {{\Lambda }^{k}{V}_{Y},{d}_{0}}\right) \otimes \left( {{\Lambda W},\bar{d}}\right) \rightarrow \left( {{\Lambda }^{k}{V}_{Y},{d}_{0}}\right) \otimes \left( {{\Lambda V},\bar{d}}\right) ,\n\]\nwhere \( {d}_{0} : \Lambda {V}_{Y} \rightarrow \Lambda {V}_{Y} \) is the ’linear part’ of \( d \) . Thus these maps are all quasi-isomorphisms.\n\nAn obvious induction on \( k \) now shows that \( \varphi \) induces quasi-isomorphisms\n\n\[ \theta \left( k\right) : \left( {\Lambda {V}_{Y} \otimes {\Lambda W}, d}\right) /I\left( k\right) \overset{ \simeq }{ \rightarrow }\left( {\Lambda {V}_{Y} \otimes {\Lambda V}, d}\right) /J\left( k\right)\n\]\n\nfor \( k \geq 1 \) . Since \( {V}_{Y} = {\left\{ {V}_{Y}^{i}\right\} }_{i \geq 1} \) by the definition of a Sullivan algebra, \( I\left( k\right) \) and \( J\left( k\right) \) are concentrated in degrees \( \geq k \) . Thus we may identify \( {H}^{i}\left( \varphi \right) = {H}^{i}\left( {\theta \left( k\right) }\right) \) for \( i < k \), and so \( \varphi \) is a quasi-isomorphism too. Since \( {m\varphi } \sim n \), so is \( n \) .
|
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\[ m : \left( {\Lambda \left( {e, u}\right) ,{du} = e}\right) \rightarrow {A}_{PL}\left( {P{S}^{k}}\right) \]\n\nby \( {me} = {A}_{PL}\left( p\right) {m}_{S}e \) and \( {mu} = \Phi \), where \( \Phi \) is any cochain satisfying \( {d\Phi } = \) \( {A}_{PL}\left( p\right) {m}_{S}e \) . (Since \( P{S}^{k} \) is contractible any cocycle of positive degree is a coboundary.) By inspection, \( m \) is a quasi-isomorphism. Hence it follows from Proposition 15.5 that \( m \) factors to yield a minimal Sullivan model\n\n\[ \bar{m} : \left( {\Lambda \left( u\right) ,0}\right) \overset{ \simeq }{ \rightarrow }{A}_{PL}\left( {\Omega {S}^{k}}\right) . \]\n\nThus \( {H}^{ * }\left( {\Omega {S}^{k};\mathbb{R}}\right) \) is the polynomial algebra on a class \( \left\lbrack u\right\rbrack \) of even degree \( k - 1 \) .\n\nIf \( k \) is even then the minimal Sullivan model for \( {S}^{k} \) has the form \( {m}_{S} \) : \( \left( {\Lambda \left( {e,{e}^{\prime }}\right), d{e}^{\prime } = {e}^{2}}\right) \overset{ \simeq }{ \rightarrow }{A}_{PL}\left( {S}^{k}\right) \) . In this case \( {A}_{PL}\left( p\right) {m}_{S} \) extends to a quasi-isomorphism\n\n\[ m : \left( {\Lambda \left( {e,{e}^{\prime }, u,{u}^{\prime }}\right) ,{du} = e, d{u}^{\prime } = {e}^{\prime } - {eu}}\right) \rightarrow {A}_{PL}\left( {P{S}^{k}}\right) . \]\n\nThus a minimal Sullivan model for \( \Omega {S}^{k} \) is given by\n\n\[ \bar{m} : \left( {\Lambda \left( {u,{u}^{\prime }}\right) ,0}\right) \overset{ \simeq }{ \rightarrow }{A}_{PL}\left( {\Omega {S}^{k}}\right) \]\n\nIn particular \( {H}^{ * }\left( {\Omega {S}^{k};\mathbb{R}}\right) \) is the tensor product of the exterior algebra on the class \( \left\lbrack u\right\rbrack \) and the polynomial algebra on the class \( \left\lbrack {u}^{\prime }\right\rbrack \) . Here \( \deg \left\lbrack u\right\rbrack = k - 1 \) and \( \deg \left\lbrack {u}^{\prime }\right\rbrack = {2k} - 2 \) .
|
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 Hurewicz theorem \( {4.19},{H}_{n}\left( {{a}_{n};\mathbb{Z}}\right) : {H}_{n}\left( {{S}^{n};\mathbb{Z}}\right) \overset{ \cong }{ \rightarrow }{H}_{n}\left( {K\left( {\pi, n}\right) ;\mathbb{Z}}\right) \) . Hence \( {H}^{n}\left( {{a}_{n};\mathbb{Q}}\right) \) is also an isomorphism. Moreover, for \( n \geq 2,\Omega {a}_{n} : \Omega {S}^{n} \rightarrow {\Omega K}\left( {\mathbb{Z}, n}\right) = \) \( K\left( {\mathbb{Z}, n - 1}\right) \) and \( {\pi }_{n - 1}\left( {\Omega {a}_{n}}\right) \) is an isomorphism, by the long exact homotopy sequence applied to the path space fibrations. Hence, as above, \( {H}^{n - 1}\left( {\Omega {a}_{n};\mathbb{Q}}\right) \) is an isomorphism.
|
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\[ \left( {{A}_{PL}\left( Y\right) \otimes \Lambda \left( e\right), d}\right) \overset{ \simeq }{ \rightarrow }{A}_{PL}\left( X\right) ,\;{de} = z \in {A}_{PL}\left( Y\right) .\n\]\n\nIf \( k \) is even then the model of \( {S}^{k} \) has the form \( \left( {\Lambda \left( {e,{e}^{\prime }}\right) ,\bar{d}{e}^{\prime } = {e}^{2}}\right) \) . Thus \( p \) has a model of the form \( \left( {{A}_{PL}\left( Y\right) \otimes \Lambda \left( {e,{e}^{\prime }}\right), d}\right) \) with \( {de} \in {A}_{PL}\left( Y\right) \) and \( d{e}^{\prime } = \) \( {e}^{2} + a \otimes e + b \) for some elements \( a, b \in {A}_{PL}\left( Y\right) \) . The condition \( {d}^{2}{e}^{\prime } = 0 \) now implies that \( {2de} = - {da} \) . Replace \( e \) by \( e + \frac{1}{2}a \) to obtain a model in which \( {de} = 0 \) and \( d{e}^{\prime } = {e}^{2} + z \), some \( z \in {A}_{PL}\left( Y\right) \) . In summary, \( {A}_{PL}\left( p\right) \) has a model of the form\n\n\[ \left( {{A}_{PL}\left( Y\right) \otimes \Lambda \left( {e,{e}^{\prime }}\right), d}\right) \overset{ \simeq }{ \rightarrow }{A}_{PL}\left( X\right) ,\;\begin{array}{l} {de} = 0 \\ d{e}^{\prime } = {e}^{2} + z, z \in {A}_{PL}\left( Y\right) . \end{array}\n\]\n\nNote that in both cases \( z \) is a cocycle in \( {A}_{PL}\left( Y\right) \) whose cohomology class \( \left\lbrack z\right\rbrack \) is determined by the fibration. In particular this class is zero if and only if \( {A}_{PL}\left( X\right) \) is weakly equivalent to \( {A}_{PL}\left( {Y \times {S}^{k}}\right) \) . However in the case of even spheres an easy calculation shows that \( {H}^{ * }\left( {X;\mathbb{k}}\right) \cong {H}^{ * }\left( {Y;\mathbb{k}}\right) \otimes {H}^{ * }\left( {S}^{k}\right) \) as \( {H}^{ * }\left( {Y;\mathbb{k}}\right) \) -modules.
|
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 must have the form \( \Lambda \left( {u, x;{dx} = {u}^{n + 1}}\right) \) . In particular, \( \mathbb{C}{P}^{n} \) is formal with cohomology algebra \( {\Lambda u}/{u}^{n + 1} \) .
|
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 \( f : I \rightarrow X \) such that \( f\left( 0\right) = f\left( 1\right) \) and this defines an inclusion \( i : {X}^{{S}^{1}} \rightarrow {X}^{I} \) .
|
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 of fibrations. Finally, the constant map \( I \rightarrow {pt} \) defines a homotopy equivalence \( {X}^{I} \leftarrow {X}^{pt} = X \) that converts the diagonal \( \Delta \) to \( q \) .\n\nWe may now apply the results above to compute a Sullivan model for \( {X}^{{S}^{1}} \) as follows. Let \( \left( {{\Lambda V}, d}\right) \) be a minimal Sullivan model for \( X \) . Then multiplication\n\n\[ \mu : \left( {{\Lambda V}, d}\right) \otimes \left( {{\Lambda V}, d}\right) \rightarrow \left( {{\Lambda V}, d}\right) \]\n\nis a Sullivan representative for \( \Delta \) as follows from Example 2, \( §{12} \) (a). Convert this to a relative Sullivan algebra \( \left( {{\Lambda V} \otimes {\Lambda V} \otimes {\Lambda W}, d}\right) \overset{ \simeq }{ \rightarrow }\left( {{\Lambda V}, d}\right) \) and then by Proposition 15.8, \( \left( {{\Lambda V}, d}\right) { \otimes }_{{\Lambda V} \otimes {\Lambda V}}\left( {{\Lambda V} \otimes {\Lambda V} \otimes {\Lambda W}, d}\right) = \left( {{\Lambda V} \otimes {\Lambda W}, d}\right) \) is a Sullivan model for \( {X}^{{S}^{1}} \) .
|
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, \[ {\nu }_{X} : {V}_{X}\overset{ \cong }{ \rightarrow }{\operatorname{Hom}}_{\mathbb{Z}}\left( {{\pi }_{ * }\left( X\right) ,\mathbb{k}}\right) \] is an isomorphism.
|
## 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 on \( k - r \) . If \( k = r \) then the Hurewicz homomorphism is an isomorphism \( {\pi }_{r}\left( X\right) \overset{ \cong }{ \rightarrow } \) \( {H}_{r}\left( {X;\mathbb{Z}}\right) \) (Theorem 4.19). On the other hand, since \( {H}^{i}\left( {X;\mathbb{R}}\right) = 0 \) for \( 1 \leq \) \( i \leq r - 1 \) the minimal model for \( X \) satisfies \( {V}_{X}^{i} = 0,1 \leq i \leq r - 1 \) (Proposition 12.2). Thus \( H\left( {m}_{X}\right) : {V}_{X}^{r}\overset{ \cong }{ \rightarrow }{H}^{r}\left( {X;\mathbb{R}}\right) \) . It is immediate from the definition that these isomorphisms identify \( {\nu }_{X} \) with the isomorphism \( {H}^{r}\left( {X;\mathbb{k}}\right) \overset{ \cong }{ \rightarrow } \) \( {\operatorname{Hom}}_{\mathbb{Z}}\left( {{H}_{r}\left( {X;\mathbb{Z}}\right) ,\mathbb{k}}\right) \) . Thus when \( k = r,{\nu }_{X} \) is an isomorphism. Suppose now that \( k > r \) . Observe first that if \( f : X \rightarrow Y \) is a weak homotopy equivalence then \( {H}^{ * }\left( {f;\mathbb{R}}\right) \) is an isomorphism. It follows that a Sullivan representative \( {\varphi }_{f} \) is a quasi-isomorphism and hence an isomorphism (Theorem 14.11). Thus \( Q\left( f\right) \), as well as \( {\pi }_{ * }\left( f\right) \), are isomorphisms. Thus by naturality, we may replace \( X \) by any space of the same weak homotopy type. In particular, we may suppose \( X \) is a CW complex. Let \( K \) be an Eilenberg-MacLane space of type \( \left( {{\pi }_{r}\left( X\right), r}\right) \) . Choose a continuous map \( g : X \rightarrow K \) such that \( {\pi }_{r}\left( g\right) = \) identity (Proposition 4.21). Factor \( g \) (as in \( §2\left( \mathrm{c}\right) \) ) in the form with \( \lambda \) a homotopy equivalence and \( p \) a fibration. Again, use naturality to replace \( X \) by \( X{ \times }_{K}{MK} \) . Thus we may assume there is a fibration \[ p : X
|
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 \[ {\pi }_{n}\left( {S}^{2k}\right) \otimes \mathbb{Q} = \left\{ \begin{array}{ll} \mathbb{Q} \cdot \iota &, n = {2k} \\ \mathbb{Q} \cdot {\left\lbrack \iota ,\iota \right\rbrack }_{W} &, n = {4k} - 1 \\ 0 & ,\text{ otherwise. } \end{array}\right. \]
|
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 13.16 gives \( \left\langle {{e}^{\prime };{\left\lbrack \iota ,\iota \right\rbrack }_{W}}\right\rangle = - \left\langle {{e}^{2};\iota ,\iota }\right\rangle = - 2 \) . Now apply Theorem 15.11
|
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 \( {\left\lbrack {a}_{0},{\left\lbrack {a}_{0},{a}_{1}\right\rbrack }_{W}\right\rbrack }_{W} \) and \( {\left\lbrack {a}_{1},{\left\lbrack {a}_{1},{a}_{0}\right\rbrack }_{W}\right\rbrack }_{W} \) . The minimal Sullivan model of \( X \) was shown to have the form \( \left( {{\Lambda V}, d}\right) = \left( {\Lambda \left( {{e}_{0},{e}_{1}, x, w,{w}_{0},\ldots }\right), d}\right) \), where \( {dx} = {e}_{0}{e}_{1},{dw} = {e}_{0}{e}_{1}x \) and the \( {w}_{i} \) have higher degree.
|
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 the Hurewicz theorem 4.19.
|
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 \( {\pi }_{1}\left( X\right) \rightarrow \) \( {\pi }_{1}\left( {X \cup \mathop{\bigcup }\limits_{i}{e}_{i}^{2}}\right) \) is surjective. Hence \( X \cup \left( {\mathop{\bigcup }\limits_{i}{e}_{i}^{2}}\right) \) is simply connected. On the other hand, since \( {H}_{1}\left( {X;\mathbb{Z}}\right) = 0 \), the long exact homology sequence for the pair \( \left( {X \cup \left( {\mathop{\bigcup }\limits_{i}{e}_{i}^{2}}\right), X}\right) \) shows that there are homology classes \( {\alpha }_{1},\ldots ,{\alpha }_{n} \in \) \( {H}_{2}\left( {X \cup \mathop{\bigcup }\limits_{i}{e}_{i}^{2};\mathbb{Z}}\right) \) that project to the classes \( \left\lbrack {e}_{1}^{2}\right\rbrack ,\ldots ,\left\lbrack {e}_{n}^{2}\right\rbrack \) in the relative homology.\n\nSince \( X \cup \left( {\mathop{\bigcup }\limits_{i}{e}_{i}^{2}}\right) \) is simply connected the Hurewicz Theorem implies that the \( {\alpha }_{i} \) can be represented by maps \( {a}_{i} : {S}^{2} \rightarrow X \cup \left( {\mathop{\bigcup }\limits_{i}{e}_{i}^{2}}\right) \) . Attach three cells by these maps to create the topological space \( Y = X \cup \left( {\mathop{\bigcup }\limits_{i}{e}_{i}^{2}}\right) \cup \left( {\mathop{\bigcup }\limits_{j}{e}_{j}^{3}}\right) \) . This is called the Quillen plus construction on \( X \) .\n\nThis construction has two important properties. Firstly \( Y \) is simply connected. Secondly the inclusion \( X \rightarrow Y \) induces an isomorphism of homology, as follows immediately from the long exact homology sequence for \( \left( {Y, X}\right) \) . In particular in the case \( X \) is an Eilenberg-MacLane space (and hence has no higher homotopy groups) the higher homotopy groups of \( Y \) are invariants of the group \( {\pi }_{1}\left( X\right) \) ; its algebraic \( K \) -groups.
|
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 Sullivan model for \( X \) (over \( \mathbb{R} \) ). In particular, if \( X \) is simply connected and \( {H}_{ * }\left( {X;\mathbb{R}}\right) \) has finite type (e.g. if \( X \) is compact) then there is an isomorphism
|
\[ {\nu }_{X} : V\overset{ \cong }{ \rightarrow }{\operatorname{Hom}}_{\mathbb{Z}}\left( {{\pi }_{ * }\left( X\right) ,\mathbb{R}}\right) \] (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\[ \left\langle {{d}_{0}{\zeta z};\alpha }\right\rangle = {\left( -1\right) }^{k + 1}\left\langle {H\left( \bar{m}\right) \left\lbrack z\right\rbrack ,{\operatorname{hur}}_{F}{\partial }_{ * }\alpha }\right\rangle .\n\]\n\nUse the pullback of the fibration via \( a \), and naturality, to reduce to the case \( Y = {S}^{k + 1} \) and \( a = i{d}_{{S}^{k + 1}} \) . Then \( \left( {\Lambda {V}_{Y}, d}\right) = \left( {\Lambda \left( {e,\ldots }\right), d}\right) \) and \( {dz} = {d}_{0}{\zeta z} = {\lambda e} \) , for some \( \lambda \in \mathbb{k} \) .\n\nNow the argument used in 15.13 for the path space fibration (with the last paragraph suppressed) applies verbatim to give the proposition.
|
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 the finitely generated polynomial algebra, \( \Lambda {V}_{{B}_{G}} \) .
|
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 \( {V}_{{B}_{G}} \cong \) \( {\operatorname{Hom}}_{\mathbb{Z}}\left( {{\pi }_{ * }\left( {B}_{G}\right) ,\mathbb{k}}\right) \) . Thus \( {\pi }_{ * }\left( {B}_{G}\right) \otimes \mathbb{Q} \) is finite dimensional and concentrated in even degrees and, for degree reasons, all the rational Whitehead products for \( {B}_{G} \) vanish.\n\nLet \( {y}_{1},\ldots ,{y}_{r} \) be the basis of \( {V}_{{B}_{G}} \) corresponding to the basis \( {x}_{1},\ldots ,{x}_{r} \) of \( {P}_{G} \) , and define a contractible Sullivan algebra \( \left( {\Lambda {V}_{{B}_{G}} \otimes \Lambda {P}_{G}, d}\right) \) by setting \( d{x}_{i} = {y}_{i} \) and \( d{y}_{i} = 0 \) . Since \( {H}^{ * }\left( {E}_{G}\right) = \mathbb{k} \), there is a commutative diagram of morphisms,\n\n\n\ndefined by setting \( m{x}_{i} = {\Phi }_{i} \), where \( {\Phi }_{i} \in {A}_{PL}\left( {E}_{G}\right) \) is any element satisfying \( d{\Phi }_{i} = {A}_{PL}\left( {p}_{G}\right) {m}_{{B}_{G}}{y}_{i} \) . Here \( m \) is a quasi-isomorphism by inspection and so this is the special case of diagram (15.4) for the universal bundle. In particular (Proposition 15.5), \( \bar{m} \) is a quasi-isomorphism. Thus we may take \( {m}_{G} = \bar{m} \) .
|
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 homogeneous space.
|
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, \( {H}^{i}\left( {\Lambda {V}_{{B}_{G}} \otimes \Lambda {V}_{X}, D}\right) = 0,\;i > \dim X - \dim G \) .
|
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 \( \dim X/G = \dim X - \dim G \), the second assertion follows.
|
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, \( {H}^{i}\left( {\Lambda {V}_{{B}_{G}} \otimes \Lambda {V}_{X}, D}\right) = 0,\;i > \dim X - \dim G \) .
|
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 \( \dim X/G = \dim X - \dim G \), the second assertion follows.
|
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 \( p \) : \( X \rightarrow {S}^{3} \times {S}^{3} \) . Let\n\n\[ q : X \rightarrow {S}^{3} \]\n\nbe the composite of \( p \) with projection on the left factor.\n\nIt is a relatively easy exercise to exhibit \( p \) as a smooth fibre bundle, and a theorem of Ehresmann then asserts that \( q \) is a smooth fibre bundle too. (It is trivial that \( q \) is a Serre fibration.) The fibre of \( q \) is the compact Lie group \( {S}^{3} \times {SU}\left( 3\right) \) . Now the discussion above gives a model for the principal bundle \( p \) , from which we find that \( q \) is represented by\n\n\[ \left( {{\Lambda u},0}\right) \rightarrow \left( {\Lambda \left( {u, v,{x}_{2},{x}_{3}}\right), d}\right) \]\n\nwith \( \deg u = \deg v = 3,{du} = {dv} = d{x}_{2} = 0 \) and \( d{x}_{3} = {uv} \) . This is not the model of a principal bundle and so \( q \) is not principal.
|
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\[ \left( {\Lambda \left( {{a}_{1},{a}_{2}}\right) \otimes \Lambda \left( {u, v,{x}_{2},{x}_{3}}\right), D}\right) \rightarrow \left( {\Lambda \left( {u, v,{x}_{2},{x}_{3}}\right), d}\right) \]\n\nwith \( \deg {a}_{i} = 4 \), and \( D{a}_{i} = 0 \) . For degree reasons \( D{x}_{3} = d{x}_{3} = {uv} \) and \( {Du},{Dv} \in \Lambda \left( {{a}_{1},{a}_{2}}\right) \) . Thus\n\n\[ 0 = {D}^{2}{x}_{3} = {Du} \otimes v - {Dv} \otimes u, \]\n\nwhence \( {Du} = {Dv} = 0 \) . We also have \( D{x}_{2} \in \Lambda \left( {{a}_{1},{a}_{2}}\right) \) and these calculations now show that \( \Lambda \left( {{a}_{1},{a}_{2}}\right) /\left( {D{x}_{2}}\right) \) is a subalgebra of \( {H}^{ * }\left( {X}_{G}\right) \) .\n\nThis subalgebra cannot be finite dimensional and hence (Proposition 15.17) the action cannot be free.
|
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 \( {H}_{ * }\left( {\Delta }_{\text{top }}\right) : {H}_{ * }\left( {{\Omega X};\mathbf{k}}\right) \rightarrow {H}_{ * }\left( {{\Omega X} \times {\Omega X};\mathbf{k}}\right) \) is a morphism of graded algebras. On the other hand \( H\left( {AW}\right) = H{\left( EZ\right) }^{-1} \) as shown in Proposition 4.10. Since \( H\left( {EZ}\right) \) is a morphism of graded algebras (Lemma 16.2) so is \( H\left( {AW}\right) \) . Hence so is \( H\left( \Delta \right) \) .
|
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 \( \alpha \in {H}_{2k}\left( {\Omega {S}^{{2k} + 1};\mathbb{k}}\right) \) . The inclusion of \( \alpha \) extends to a unique morphism from the polynomial algebra \( \mathbb{k}\left\lbrack \alpha \right\rbrack \) . We show this is an isomorphism:\n\n\[ \mathbb{k}\left\lbrack \alpha \right\rbrack \overset{ \cong }{ \rightarrow }{H}_{ * }\left( {\Omega {S}^{{2k} + 1};\mathbb{k}}\right) \]\n\nIn fact \( \left\lbrack {S}^{2k}\right\rbrack \) is (trivially) primitive in \( {H}_{ * }\left( {{S}^{2k};\mathbb{R}}\right) \) and so \( \alpha \) is primitive in \( {H}_{ * }\left( {\Omega {S}^{{2k} + 1};\mathbb{k}}\right) \) . Since \( H\left( \Delta \right) \) is an algebra morphism, and since \( k \) is even,\n\n\[ H\left( \Delta \right) {\alpha }^{n} = {\left( \alpha \otimes 1 + 1 \otimes \alpha \right) }^{n} = {\alpha }^{n} \otimes 1 + n{\alpha }^{n - 1} \otimes \alpha + \cdots + 1 \otimes {\alpha }^{n},\;n \geq 1. \]\n\nIf \( {\alpha }^{n - 1} \neq 0 \) then it follows that \( H\left( \Delta \right) {\alpha }^{n} \neq 0 \) and so \( {\alpha }^{n} \neq 0 \) . Hence our morphism is injective. But in Example 1, \( §{15} \) (b) we computed the Hilbert series of \( {H}^{ * }\left( {\Omega {S}^{{2k} + 1};\mathbb{k}}\right) \) to be \( {\left( 1 - {z}^{2k}\right) }^{-1} \) . Thus\n\n\[ \dim {H}_{i}\left( {\Omega {S}^{{2k} + 1};\mathbb{k}}\right) = \dim {H}^{i}\left( {\Omega {S}^{{2k} + 1};\mathbb{k}}\right) = \left\{ {\begin{array}{ll} 1 &, i = {2rk} \\ 0 & ,\text{ otherwise } \end{array} = \dim \mathbb{k}{\left\lbrack \alpha \right\rbrack }_{i}.}\right. \]\n\nIt follows that the morphism above is an isomorphism. At the end of \( §{16}\left( \mathrm{\;d}\right) \) we establish the same result for \( {H}_{ * }\left( {\Omega {S}^{2k};\mathbb{k}}\right) \) .
|
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 \( {\Lambda }^{2}\left( {{V}_{X} \oplus V}\right) \) is zero, and this fact translates to\n\n\[ \n{d}_{1}{d}_{0}v = - \mathop{\sum }\limits_{i}{\left( -1\right) }^{\deg {u}_{i}}{u}_{i} \land {d}_{0}{v}_{i}\n\]\n\nOn the other hand, since \( {\varphi dv} = \left( {d \otimes {id}}\right) {\varphi v} \), the components of \( {\varphi dv} \) and \( \left( {d \otimes {id}}\right) {\varphi v} \) in \( {V}_{X} \otimes 1 \otimes V \) also coincide. Use the following facts to compute these components:\n\n- \( \operatorname{Im}d \subset {\Lambda }^{ + }{V}_{X} \otimes {\Lambda V} \bullet \operatorname{Im}\left( {\varphi - {id} \otimes \bar{\varphi }}\right) \subset {\Lambda }^{ + }{V}_{X} \otimes {\Lambda V} \otimes {\Lambda V} \)\n\n\[ \n\text{-}\varphi = {id}\text{in}\Lambda {V}_{X}\; \bullet \bar{\varphi }{v}_{i} - 1 \otimes {v}_{i} \in {\Lambda }^{ + }V \otimes {\Lambda V}\text{.\n}\]\n\nThis gives the equation \( \sum {u}_{i} \otimes 1 \otimes {v}_{i} = \sum {d}_{0}\zeta {\Phi }_{i} \otimes 1 \otimes \zeta {\Psi }_{i} \), whence \( \mathop{\sum }\limits_{i}{\left( -1\right) }^{\deg {u}_{i}}{u}_{i} \land \)\n\n\( {d}_{0}{v}_{i} = - \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
|
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 \( {H}_{ * }\left( \overline{id}\right) \left\lbrack {S}^{2{k}_{j}}\right\rbrack \) simply by \( \left\lbrack {S}_{j}\right\rbrack \in {H}_{ * }\left( {\Omega {S}^{2{k}_{j} + 1};\mathbb{k}}\right) \) . The classes \( 1,\left\lbrack {S}_{j}\right\rbrack ,{\left\lbrack {S}_{j}\right\rbrack }^{2},\ldots \) are a basis for \( {H}_{ * }\left( {\Omega {S}^{2{k}_{j} + 1};\mathbb{k}}\right) \), as we showed in the Example at the start of \( §{16} \) . Denote \( \left\lbrack {S}^{2{\ell }_{i} + 1}\right\rbrack \) simply by \( \left\lbrack {S}_{i}\right\rbrack \), so that 1 and \( \left\lbrack {S}_{i}\right\rbrack \) are a basis for \( {H}_{ * }\left( {{S}^{2{\ell }_{i} + 1};\mathbb{k}}\right) \) . Then (Eilenberg-Zilber) the classes\n\n\[ \left\lbrack {S}_{{i}_{1}}\right\rbrack \otimes \cdots \otimes \left\lbrack {S}_{{i}_{p}}\right\rbrack \otimes {\left\lbrack {S}_{{j}_{1}}\right\rbrack }^{{n}_{1}} \otimes \cdots \otimes {\left\lbrack {S}_{{j}_{q}}\right\rbrack }^{{n}_{q}} \]\n\nare a basis for \( {H}_{ * }\left( {\mathop{\prod }\limits_{i}^{ \sim }{S}^{2{\ell }_{i} + 1} \times \mathop{\prod }\limits_{j}^{ \sim }\Omega {S}^{2{k}_{j} + 1};\mathbb{R}}\right) \) . This basis is mapped by \( {H}_{ * }\left( f\right) \) onto the set of distinguished homology classes in \( {H}_{ * }\left( {{\Omega X};\mathbb{R}}\right) \) .
|
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 \( \leq 2\ell + 1 \) . The homology of \( {S}^{2\ell + 1} \) vanishes in higher degrees (4.14) as does \( {\pi }_{ * }\left( {S}^{2\ell + 1}\right) \otimes \mathbb{R} \) (Example 1, \( §{15}\left( \mathrm{\;d}\right) \) ).\n\n(ii) hur: \( {\pi }_{ * }\left( {\Omega {S}^{{2k} + 1}}\right) \otimes \mathbb{R}\overset{ \cong }{ \rightarrow }{P}_{ * }\left( {\Omega {S}^{{2k} + 1};\mathbb{R}}\right), k \geq 1 \) .\n\nSince \( {\pi }_{ * }\left( {\Omega {S}^{{2k} + 1}}\right) \otimes \mathbf{k} \cong {\pi }_{* + 1}\left( {S}^{{2k} + 1}\right) \otimes \mathbf{k} \) we conclude, as in (i) that this graded vector space is concentrated in degree \( {2k} \) . Thus the Hurewicz Theorem 4.19 asserts that hur: \( {\pi }_{ * }\left( {\Omega {S}^{{2k} + 1}}\right) \otimes \mathbf{k}\overset{ \simeq }{ \rightarrow }{P}_{2k}\left( {\Omega {S}^{{2k} + 1};\mathbf{k}}\right) \) . In the Example at the start of \( §{16} \) we saw that \( {H}_{ * }\left( {\Omega {S}^{{2k} + 1};\mathbb{R}}\right) = \mathbb{R}\left\lbrack \alpha \right\rbrack \) with \( \deg \alpha = {2k} \), and that \( H\left( \Delta \right) {\alpha }^{n} = \mathop{\sum }\limits_{{i = 0}}^{n}\left( \begin{matrix} n \\ i \end{matrix}\right) {\alpha }^{i} \otimes {\alpha }^{n - i} \) . This shows that \( {\alpha }^{n} \) is not primitive for \( n \geq 2 \) , so that \( {P}_{ * }\left( {\Omega {S}^{{2k} + 1};\mathbb{k}}\right) = {P}_{2k}\left( {\Omega {S}^{{2k} + 1};\mathbb{k}}\right) \).\n\n(iii) If \( Y \) and \( Z \) are path connected topological spaces, then\n\n\[{\pi }_{ * }\left( {Y \times Z}\right) = {\pi }_{ * }\left( Y\right) \oplus {\pi }_{ * }\left( Z\right) \;\text{ and }\;{P}_{ * }\left( {Y \times Z;\mathbb{R}}\right) = {P}_{ * }\left( {Y;\mathbb{R}}\right) \oplus {P}_{ * }\left( {Z;\mathbb{R}}\right) .\n\]\n\nThe first assertion is immediate from the definition of \( {\pi }_{ * } \) . For the second, recall that Eilenberg-Zilber induces an isomorphism \( {H}_{ * }\left( {Y;\mathbb{R}}\right) \otimes {H}_{ * }\left( {Z;\mathbb{R}}\right) \overset{ \cong }{ \rightarrow }{H}_{ * }(Y \times \)
|
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\[ \left\lbrack {\theta {\gamma }_{0},\theta {\gamma }_{1}}\right\rbrack = {\left( -1\right) }^{k + 1}\theta \left( {\left\lbrack {\gamma }_{0},{\gamma }_{1}\right\rbrack }_{W}\right) . \]
|
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 \( \bar{\varphi }v = v \otimes 1 + \sum {\Phi }_{i} \otimes {\Psi }_{i} + 1 \otimes v \) . Since \( {d}_{0}\zeta \) is dual to \( \theta \) (Proposition 15.14) and since \( \bar{\varphi } \) is dual to the multiplication in \( {H}_{ * }\left( {{\Omega X};\mathbb{R}}\right) \), we have\n\n\[ \left\langle {H\left( \bar{m}\right) v,\left\lbrack {\theta {\gamma }_{0},\theta {\gamma }_{1}}\right\rbrack }\right\rangle = \left\langle {H\left( \bar{m}\right) v,\left( {\theta {\gamma }_{0}}\right) \left( {\theta {\gamma }_{1}}\right) - {\left( -1\right) }^{kn}\left( {\theta {\gamma }_{1}}\right) \left( {\theta {\gamma }_{0}}\right) }\right\rangle \]\n\n\[ = \left\langle {{\sum H}\left( \bar{m}\right) {\Phi }_{i} \otimes H\left( \bar{m}\right) {\Psi }_{i},\theta {\gamma }_{0} \otimes \theta {\gamma }_{1} - {\left( -1\right) }^{kn}\theta {\gamma }_{1} \otimes \theta {\gamma }_{0}}\right\rangle \]\n\n\[ = \mathop{\sum }\limits_{i}{\left( -1\right) }^{k\deg {\Psi }_{i}}\left\langle {H\left( \bar{m}\right) {\Phi }_{i},\theta {\gamma }_{0}}\right\rangle \left\langle {H\left( \bar{m}\right) {\Psi }_{i},\theta {\gamma }_{1}}\right\rangle \]\n\n\[ - \mathop{\sum }\limits_{i}{\left( -1\right) }^{{kn} + n\deg {\Psi }_{i}}\left\langle {H\left( \bar{m}\right) {\Phi }_{i},\theta {\gamma }_{1}}\right\rangle \left\langle {H\left( \bar{m}\right) {\Psi }_{i},\theta {\gamma }_{0}}\right\rangle \]\n\n\[ = \mathop{\sum }\limits_{i}{\left( -1\right) }^{{kn} + k + n}\left\langle {{d}_{0}\zeta {\Phi }_{i};{\gamma }_{0}}\right\rangle \left\langle {{d}_{0}\zeta {\Psi }_{i};{\gamma }_{1}}\right\rangle \]\n\n\[ + \mathop{\sum }\limits_{i}{\left( -1\right) }^{k + n + 1}\left\langle {{d}_{0}\zeta {\Phi }_{i};{\gamma }_{1}}\right\rangle \left\langle {{d}_{0}\zeta {\Psi }_{i};{\gamma }_{0}}\right\rangle \]\n\n\[ = {\left( -1\right) }^{k + 1}\left\langle {\mathop{\sum }\limits_{i}{\left( -1\right) }^{\deg {\Phi }_{i}}\left( {{d}_{0}\zeta {\Phi }_{i}}\right) \land \left( {{d}_{0}\zeta {\Psi }_{i}}\right) ;{\gamma }_{0},{\gamma }_{1}}\right\rangle \]\n\nOn the other hand, we may apply Proposition 13.16 where we calculated the Whitehead product to obtain\n\n\[ \left\langle {H\left( \bar{m}\right) v;\theta {\left\lbrack {\gamma }_{0},{\gamma }_{1}\right\rbrack }_{W}}\right\rangle = {\left( -1\right) }^{k + n + 1}\left\langle {{d}_{0}v;{\left\lbrack {\gamma }_{0},{\gamma }_{1}\right\rbrack }_{W}}\right\rangle \]\n\n\[ = \left\langle {{d}_{1}{d}_{0}v;{\gamma }_{0},{\gamma }_{1}}\right\rangle \]\n\nThe Proposition follows from these formulae, given the duality between \( V \) and \( {P}_{ * }\left( {{\Omega X};\mathbb{k}}\right) \) established in Lemma 16.9.
|
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 identifies \( {hu}{r}_{\Omega X} \) as a linear map \( {L}_{X} \rightarrow {H}_{ * }\left( {{\Omega X};\mathbb{R}}\right) \), which then automatically extends to a morphism \( T{L}_{X} \rightarrow \) \( {H}_{ * }\left( {{\Omega X};\mathbb{R}}\right) \) of graded algebras. Proposition 16.11 asserts precisely that the elements (16.12) are in the kernel of this morphism. Hence it factors over the quotient \( T{L}_{X} \rightarrow \left( {T{L}_{X}}\right) /I \) to define a morphism of graded algebras \[ \sigma : \left( {T{L}_{X}}\right) /I \rightarrow {H}_{ * }\left( {{\Omega X};\mathbb{k}}\right) \] We have to show that \( \sigma \) is an isomorphism. As in \( §{16}\left( \mathrm{\;b}\right) \) let \( {b}_{i} : {S}^{2{\ell }_{i} + 2} \rightarrow X \) and \( {a}_{j} : {S}^{2{k}_{j} + 1} \rightarrow X \) represent a basis of \( {\pi }_{ * }\left( X\right) \otimes \mathbf{k} \) . Put \( {y}_{i} = {s}^{-1}\left\lbrack {b}_{i}\right\rbrack \) and \( {x}_{j} = {s}^{-1}\left\lbrack {a}_{j}\right\rbrack \) . Thus \( \left\{ {{y}_{i},{x}_{j}}\right\} \) is a basis of \( {L}_{X} \) . Recall that \( {T}^{s}{L}_{X} = {L}_{X} \otimes \cdots \otimes {L}_{X} \) ( \( s \) factors). Let \( {F}_{s} \subset \left( {T{L}_{X}}\right) /I \) be the image of \( {T}^{ \leq s}{L}_{X} \) ; elements in \( {F}_{s} \) have filtration degree \( s \) . By the very definition of \( I \), if \( z, w, \in {L}_{X} \) then \( {zw} - {\left( -1\right) }^{\deg z\deg w}{wz} \) has filtration length 1 . In particular if \( z \) has odd degree then \( {z}^{2} = \frac{1}{2}\left( {{zz} + {zz}}\right) \) has filtration degree 1 . An obvious induction on filtration degree now shows that every element in \( \left( {T{L}_{X}}\right) /I \) is a linear combination of the elements \[ {i}_{1} < \cdots < {i}_{p} \] \[ {y}_{{i}_{1}}\cdots \cdots {y}_{{i}_{p}} \cdot {x}_{{j}_{1}}^{{n}_{1}}\cdots \cdots {x}_{{j}_{q}}^{{n}_{q}},{j}_{1} < \cdots < {j}_{q} \] \[ {n}_{i} \in \mathbb{N}\text{.} \] (Show that the subset with \( p + \sum {n}_{\ell } \leq s \) spans \( {F}^{s} \) .) The morphism \( \sigma \) maps these elements to the distinguished basis of \( {H}_{ * }\left( {{\Omega X};\mathbb{R}}\right) \) established in Proposition 16.8. Hence the elements above must be a basis of \( \left( {T{L}_{X}}\right) /I \) and \( \sigma \) must be an isomorphism.
|
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 \( \alpha = \operatorname{hur}\left\lbrack a\right\rbrack = {H}_{ * }\left( a\right) \left\lbrack {S}^{n}\right\rbrack \in \) \( {H}_{n}\left( {\Omega {S}^{n};\mathbb{R}}\right) \) . We show that the inclusion of \( \alpha \) extends to an isomorphism\n\n\[ \mathbb{k}\left\lbrack \alpha \right\rbrack \overset{ \cong }{ \rightarrow }{H}_{ * }\left( {\Omega {S}^{n + 1};\mathbb{k}}\right) \]\n\nof graded algebras. If \( n \) is odd this has already been done in the example at the start of \( §{16} \) .\n\nSuppose \( n = {2k} \) . Let \( \iota = \left\lbrack {i{d}_{{S}^{2n}}}\right\rbrack \in {\pi }_{2k}\left( {S}^{2k}\right) \) . Then \( {\pi }_{ * }\left( {S}^{2k}\right) \otimes \mathbb{k} \) is two dimensional, with basis given by \( \iota \) and \( {\left\lbrack \iota ,\iota \right\rbrack }_{W} \) (Example 1, \( §{15}\left( \mathrm{\;d}\right) \) ). Apply Theorem 16.13 to obtain an isomorphism\n\n\[ T\left( {x, y}\right) /\left( {{x}^{2} - \frac{1}{2}y}\right) \overset{ \cong }{ \rightarrow }{H}_{ * }\left( {\Omega {S}^{2k};\mathbb{k}}\right) .\n\nThen note that \( T\left( {x, y}\right) /\left( {{x}^{2} - \frac{1}{2}y}\right) = T\left( x\right) = \mathbb{R}\left\lbrack x\right\rbrack \), and that this isomorphism sends \( x \mapsto \alpha \) .
|
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 degenerate (if \( \left. {\tau = {s}_{j}\omega \text{ then }\tau \times y \sim \omega \times {\varrho }_{j}y}\right) \) and \( y \in {\overset{ \circ }{\Delta }}^{k} \) (if \( y = {\lambda }_{i}z \) then \( \tau \times y \sim {\partial }_{i}\tau \times z \) ). Hence \( \widetilde{q} \) is surjective. A tedious computation using the commutation formulae (10.2) shows that \( \widetilde{q} \) is injective.
|
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 \( \{ \sigma \} \times \partial {\Delta }^{n} \) .
|
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 continuous map \( {q}_{\sigma } \) : \n\n\( \{ \sigma \} \times \partial {\Delta }^{n} \rightarrow \left| {K\left( {n - 1}\right) }\right| \) . Thus, because of Lemma 17.1, \( q \) induces a continuous bijection \n\n\[ \nq\left( n\right) : \left| {K\left( {n - 1}\right) }\right| { \cup }_{\left( {q}_{\sigma }\right) }\left( {\mathop{\coprod }\limits_{{\sigma \in N{K}_{n}}}\{ \sigma \} \times {\Delta }^{n}}\right) \rightarrow \left| {K\left( n\right) }\right| . \n\] \n\nA straightforward check shows that the \( q\left( n\right) \) are proper, and hence homeomorphisms. Thus the \( q\left( n\right) \) define a continuous bijection to \( \left| K\right| \) from a CW complex with the desired properties. This is also proper, and hence a homeomorphism.
|
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 \[ {\left( CK\right) }_{n} = {K}_{n} \coprod \left( {\mathop{\coprod }\limits_{{k = 1}}^{{n - 1}}{K}_{k} \times \left\{ {c}_{n - k - 1}\right\} }\right) \coprod \left\{ {c}_{n}\right\} \] by setting (for \( \sigma \in {K}_{k} \) ): \[ {\partial }_{i}\left( {\sigma ,{c}_{r}}\right) = \left\{ \begin{array}{ll} \left( {{\partial }_{i}\sigma ,{c}_{r}}\right) &, i \leq k. \\ \sigma & ,\text{ if }r = 0\text{ and }i = k + 1. \\ \left( {\sigma ,{c}_{r - 1}}\right) & ,\text{ otherwise. } \end{array}\right. \] and \[ {s}_{j}\left( {\sigma ,{c}_{r}}\right) = \left\{ \begin{array}{ll} \left( {{s}_{j}\sigma ,{c}_{r}}\right) &, j \leq k. \\ \left( {\sigma ,{c}_{r + 1}}\right) & ,\text{ otherwise. } \end{array}\right. \] This defines a simplicial set \( {CK} \), the cone on \( K \) .
|
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 applies verbatim to simplicial sets.) In particular the identity of \( K \) extends to a morphism \( \psi : {CK} \rightarrow K \) . Thus \( i{d}_{\left| K\right| } = \left| \psi \right| \circ \left| \lambda \right| \) where \( \lambda : K \rightarrow {CK} \) is the inclusion. Since \( \left| {CK}\right| \) is contractible, \( i{d}_{\left| K\right| } \sim \) constant map; i.e. \( \left| K\right| \) is contractible.
|
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 (cf. Example 1 of \( §{17}\left( \mathrm{a}\right) \) ) that\n\n\[ \n{q}_{\sigma } = \left| {\sigma }_{ * }\right| : \left| {\Delta \left\lbrack n\right\rbrack }\right| \rightarrow \left| L\right| .\n\]\n\nThus the pullback of \( \left| \varrho \right| \) to an \( n \) -cell is just the fibre product \( \left| {\Delta \left\lbrack n\right\rbrack }\right| { \times }_{\left| L\right| }\left| K\right| \) defined with respect to \( \left| {\sigma }_{ * }\right| : \left| {\Delta \left\lbrack n\right\rbrack }\right| \rightarrow \left| L\right| \) and \( \left| \varrho \right| : \left| K\right| \rightarrow \left| L\right| .\n\nNow apply Proposition 17.4 to translate the simplicial diagrams above to the commutative topological diagrams\n\n\n\n\n\nThese show that \( \left| \varrho \right| \) pulls back to a product bundle with fibre \( \left| F\right| \) over every cell. Thus Proposition 2.7 asserts that \( \left| \varrho \right| \) is a fibre bundle with fibre \( \left| F\right| \) .
|
Yes
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.