68 (where $X$ is a component of a permissible decomposition of $Y$) to get a blob diagram on |
68 (where $X$ is a component of a permissible decomposition of $Y$) to get a blob diagram on |
69 $Y\times F$. |
69 $Y\times F$. |
70 In filtration degrees 1 and higher we define the map to be zero. |
70 In filtration degrees 1 and higher we define the map to be zero. |
71 It is easy to check that this is a chain map. |
71 It is easy to check that this is a chain map. |
72 |
72 |
73 Next we define a map |
73 In the other direction, we will define a subcomplex $G_*\sub \bc_*^C(Y\times F)$ |
74 \[ |
74 and a map |
75 \phi: \bc_*^C(Y\times F) \to \bc_*^\cF(Y) . |
75 \[ |
76 \] |
76 \phi: G_* \to \bc_*^\cF(Y) . |
77 Actually, we will define it on the homotopy equivalent subcomplex |
77 \] |
78 $\cS_* \sub \bc_*^C(Y\times F)$ generated by blob diagrams which are small with |
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79 respect to some open cover |
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80 of $Y\times F$ |
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81 (Proposition \ref{thm:small-blobs}). |
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82 We will have to show eventually that this is independent (up to homotopy) of the choice of cover. |
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83 Also, for a fixed choice of cover we will only be able to define the map for blob degree less than |
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84 some bound, but this bound goes to infinity as the cover become finer. |
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85 |
78 |
86 Given a decomposition $K$ of $Y$ into $k$-balls $X_i$, let $K\times F$ denote the corresponding |
79 Given a decomposition $K$ of $Y$ into $k$-balls $X_i$, let $K\times F$ denote the corresponding |
87 decomposition of $Y\times F$ into the pieces $X_i\times F$. |
80 decomposition of $Y\times F$ into the pieces $X_i\times F$. |
88 |
81 |
89 %We will define $\phi$ inductively, starting at blob degree 0. |
82 Let $G_*\sub \bc_*^C(Y\times F)$ be the subcomplex generated by blob diagrams $a$ such that there |
90 %Given a 0-blob diagram $x$ on $Y\times F$, we can choose a decomposition $K$ of $Y$ |
83 exists a decomposition $K$ of $Y$ such that $a$ splits along $K\times F$. |
91 %such that $x$ is splittable with respect to $K\times F$. |
84 It follows from Proposition \ref{thm:small-blobs} that $\bc_*^C(Y\times F)$ is homotopic to a subcomplex of $G_*$. |
92 %This defines a filtration degree 0 element of $\bc_*^\cF(Y)$ |
85 (If the blobs of $a$ are small with respect to a sufficiently fine cover then their |
93 |
86 projections to $Y$ are contained in some disjoint union of balls.) |
94 We will define $\phi$ using a variant of the method of acyclic models. |
87 Note that the image of $\psi$ is contained in $G_*$. |
95 Let $a\in \cS_m$ be a blob diagram on $Y\times F$. |
88 (In fact, equal to $G_*$.) |
96 For $m$ sufficiently small there exists a decomposition $K$ of $Y$ into $k$-balls such that the |
89 |
97 codimension 1 cells of $K\times F$ miss the blobs of $a$, and more generally such that $a$ is splittable along (the codimension-1 part of) $K\times F$. |
90 We will define $\phi: G_* \to \bc_*^\cF(Y)$ using the method of acyclic models. |
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91 Let $a$ be a generator of $G_*$. |
98 Let $D(a)$ denote the subcomplex of $\bc_*^\cF(Y)$ generated by all $(a, \ol{K})$ |
92 Let $D(a)$ denote the subcomplex of $\bc_*^\cF(Y)$ generated by all $(a, \ol{K})$ |
99 such that each $K_i$ has the aforementioned splittable property. |
93 such that $a$ splits along each $K_i\times F$. |
100 (Recall that $\ol{K} = (K_0,\ldots,K_l)$ denotes a chain of decompositions; |
94 (Recall that $\ol{K} = (K_0,\ldots,K_l)$ denotes a chain of decompositions; |
101 see Subsection \ref{ss:ncat_fields}.) |
95 see Subsection \ref{ss:ncat_fields}.) |
102 \nn{need to define $D(a)$ more clearly; also includes $(b_j, \ol{K})$ where |
96 \nn{need to define $D(a)$ more clearly; also includes $(b_j, \ol{K})$ where |
103 $\bd(a) = \sum b_j$.} |
97 $\bd(a) = \sum b_j$.} |
104 (By $(a, \ol{K})$ we really mean $(a^\sharp, \ol{K})$, where $a^\sharp$ is |
98 (By $(a, \ol{K})$ we really mean $(a^\sharp, \ol{K})$, where $a^\sharp$ is |
200 |
194 |
201 Continuing in this way we see that $D(a)$ is acyclic. |
195 Continuing in this way we see that $D(a)$ is acyclic. |
202 \end{proof} |
196 \end{proof} |
203 |
197 |
204 We are now in a position to apply the method of acyclic models to get a map |
198 We are now in a position to apply the method of acyclic models to get a map |
205 $\phi:\cS_* \to \bc_*^\cF(Y)$. |
199 $\phi:G_* \to \bc_*^\cF(Y)$. |
206 This map is defined in sufficiently low degrees, sends a blob diagram $a$ to $D(a)$, |
200 We may assume that $\phi(a)$ has the form $(a, K) + r$, where $(a, K)$ is in filtration degree zero |
207 and is well-defined up to (iterated) homotopy. |
201 and $r$ has filtration degree greater than zero. |
208 |
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209 The subcomplex $\cS_* \subset \bc_*^C(Y\times F)$ depends on choice of cover of $Y\times F$. |
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210 If we refine that cover, we get a complex $\cS'_* \subset \cS_*$ |
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211 and a map $\phi':\cS'_* \to \bc_*^\cF(Y)$. |
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212 $\phi'$ is defined only on homological degrees below some bound, but this bound is higher than |
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213 the corresponding bound for $\phi$. |
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214 We must show that $\phi$ and $\phi'$ agree, up to homotopy, |
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215 on the intersection of the subcomplexes on which they are defined. |
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216 This is clear, since the acyclic subcomplexes $D(a)$ above used in the definition of |
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217 $\phi$ and $\phi'$ do not depend on the choice of cover. |
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218 |
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219 %\nn{need to say (and justify) that we now have a map $\phi$ indep of choice of cover} |
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220 |
202 |
221 We now show that $\phi\circ\psi$ and $\psi\circ\phi$ are homotopic to the identity. |
203 We now show that $\phi\circ\psi$ and $\psi\circ\phi$ are homotopic to the identity. |
222 |
204 |
223 $\psi\circ\phi$ is the identity on the nose. |
205 $\psi\circ\phi$ is the identity on the nose: |
224 $\phi$ takes a blob diagram $a$ and chops it into pieces |
206 \[ |
225 according to some decomposition $K$ of $Y$. |
207 \psi(\phi(a)) = \psi((a,K)) + \psi(r) = a + 0. |
226 $\psi$ glues those pieces back together, yielding the same $a$ we started with. |
208 \] |
227 |
209 Roughly speaking, $(a, K)$ is just $a$ chopped up into little pieces, and |
228 $\phi\circ\psi$ is the identity up to homotopy by another MoAM argument.... |
210 $\psi$ glues those pieces back together, yielding $a$. |
229 |
211 We have $\psi(r) = 0$ since $\psi$ is zero in positive filtration degrees. |
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212 |
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213 $\phi\circ\psi$ is the identity up to homotopy by another MoAM argument. |
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214 To each generator $(a, \ol{K})$ of we associated the acyclic subcomplex $D(a)$ defined above. |
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215 Both the identity map and $\phi\circ\psi$ are compatible with this |
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216 collection of acyclic subcomplexes, so by the usual MoAM argument these two maps |
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217 are homotopic. |
230 |
218 |
231 This concludes the proof of Theorem \ref{product_thm}. |
219 This concludes the proof of Theorem \ref{product_thm}. |
232 \end{proof} |
220 \end{proof} |
233 |
221 |
234 \nn{need to say something about dim $< n$ above} |
222 \nn{need to say something about dim $< n$ above} |