62 |
62 |
63 First we define a map |
63 First we define a map |
64 \[ |
64 \[ |
65 \psi: \bc_*^\cF(Y) \to \bc_*^C(Y\times F) . |
65 \psi: \bc_*^\cF(Y) \to \bc_*^C(Y\times F) . |
66 \] |
66 \] |
67 In filtration degree 0 we just glue together the various blob diagrams on $X\times F$ |
67 In filtration degree 0 we just glue together the various blob diagrams on $X_i\times F$ |
68 (where $X$ is a component of a permissible decomposition of $Y$) to get a blob diagram on |
68 (where $X_i$ 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 In the other direction, we will define a subcomplex $G_*\sub \bc_*^C(Y\times F)$ |
73 In the other direction, we will define a subcomplex $G_*\sub \bc_*^C(Y\times F)$ |
82 Let $G_*\sub \bc_*^C(Y\times F)$ be the subcomplex generated by blob diagrams $a$ such that there |
82 Let $G_*\sub \bc_*^C(Y\times F)$ be the subcomplex generated by blob diagrams $a$ such that there |
83 exists a decomposition $K$ of $Y$ such that $a$ splits along $K\times F$. |
83 exists a decomposition $K$ of $Y$ such that $a$ splits along $K\times F$. |
84 It follows from Proposition \ref{thm:small-blobs} that $\bc_*^C(Y\times F)$ is homotopic to a subcomplex of $G_*$. |
84 It follows from Proposition \ref{thm:small-blobs} that $\bc_*^C(Y\times F)$ is homotopic to a subcomplex of $G_*$. |
85 (If the blobs of $a$ are small with respect to a sufficiently fine cover then their |
85 (If the blobs of $a$ are small with respect to a sufficiently fine cover then their |
86 projections to $Y$ are contained in some disjoint union of balls.) |
86 projections to $Y$ are contained in some disjoint union of balls.) |
87 Note that the image of $\psi$ is contained in $G_*$. |
87 Note that the image of $\psi$ is equal to $G_*$. |
88 (In fact, equal to $G_*$.) |
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89 |
88 |
90 We will define $\phi: G_* \to \bc_*^\cF(Y)$ using the method of acyclic models. |
89 We will define $\phi: G_* \to \bc_*^\cF(Y)$ using the method of acyclic models. |
91 Let $a$ be a generator of $G_*$. |
90 Let $a$ be a generator of $G_*$. |
92 Let $D(a)$ denote the subcomplex of $\bc_*^\cF(Y)$ generated by all $(a, \ol{K})$ |
91 Let $D(a)$ denote the subcomplex of $\bc_*^\cF(Y)$ generated by all $(b, \ol{K})$ |
93 such that $a$ splits along each $K_i\times F$. |
92 such that $a$ splits along $K_0\times F$ and $b$ is a generator appearing |
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93 in an iterated boundary of $a$ (this includes $a$ itself). |
94 (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; |
95 see Subsection \ref{ss:ncat_fields}.) |
95 see Subsection \ref{ss:ncat_fields}.) |
96 \nn{need to define $D(a)$ more clearly; also includes $(b_j, \ol{K})$ where |
96 By $(b, \ol{K})$ we really mean $(b^\sharp, \ol{K})$, where $b^\sharp$ is |
97 $\bd(a) = \sum b_j$.} |
97 $b$ split according to $K_0\times F$. |
98 (By $(a, \ol{K})$ we really mean $(a^\sharp, \ol{K})$, where $a^\sharp$ is |
98 To simplify notation we will just write plain $b$ instead of $b^\sharp$. |
99 $a$ split according to $K_0\times F$. |
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100 To simplify notation we will just write plain $a$ instead of $a^\sharp$.) |
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101 Roughly speaking, $D(a)$ consists of filtration degree 0 stuff which glues up to give |
99 Roughly speaking, $D(a)$ consists of filtration degree 0 stuff which glues up to give |
102 $a$, filtration degree 1 stuff which makes all of the filtration degree 0 stuff homologous, |
100 $a$ (or one of its iterated boundaries), filtration degree 1 stuff which makes all of the filtration degree 0 stuff homologous, |
103 filtration degree 2 stuff which kills the homology created by the |
101 filtration degree 2 stuff which kills the homology created by the |
104 filtration degree 1 stuff, and so on. |
102 filtration degree 1 stuff, and so on. |
105 More formally, |
103 More formally, |
106 |
104 |
107 \begin{lemma} |
105 \begin{lemma} |
146 Consider a different choice of decomposition $L'$ in place of $L$ above. |
144 Consider a different choice of decomposition $L'$ in place of $L$ above. |
147 This leads to a cycle consisting of filtration degree 1 stuff. |
145 This leads to a cycle consisting of filtration degree 1 stuff. |
148 We want to show that this cycle bounds a chain of filtration degree 2 stuff. |
146 We want to show that this cycle bounds a chain of filtration degree 2 stuff. |
149 Choose a decomposition $M$ which has common refinements with each of |
147 Choose a decomposition $M$ which has common refinements with each of |
150 $K$, $KL$, $L$, $K'L$, $K'$, $K'L'$, $L'$ and $KL'$. |
148 $K$, $KL$, $L$, $K'L$, $K'$, $K'L'$, $L'$ and $KL'$. |
151 \nn{need to also require that $KLM$ antirefines to $KM$, etc.} |
149 (We also also require that $KLM$ antirefines to $KM$, etc.) |
152 Then we have a filtration degree 2 chain, as shown in Figure \ref{zzz5}, which does the trick. |
150 Then we have a filtration degree 2 chain, as shown in Figure \ref{zzz5}, which does the trick. |
153 (Each small triangle in Figure \ref{zzz5} can be filled with a filtration degree 2 chain.) |
151 (Each small triangle in Figure \ref{zzz5} can be filled with a filtration degree 2 chain.) |
154 |
152 |
155 \begin{figure}[!ht] |
153 \begin{figure}[!ht] |
156 %\begin{equation*} |
154 %\begin{equation*} |
209 Roughly speaking, $(a, K)$ is just $a$ chopped up into little pieces, and |
207 Roughly speaking, $(a, K)$ is just $a$ chopped up into little pieces, and |
210 $\psi$ glues those pieces back together, yielding $a$. |
208 $\psi$ glues those pieces back together, yielding $a$. |
211 We have $\psi(r) = 0$ since $\psi$ is zero in positive filtration degrees. |
209 We have $\psi(r) = 0$ since $\psi$ is zero in positive filtration degrees. |
212 |
210 |
213 $\phi\circ\psi$ is the identity up to homotopy by another MoAM argument. |
211 $\phi\circ\psi$ is the identity up to homotopy by another MoAM argument. |
214 To each generator $(a, \ol{K})$ of we associated the acyclic subcomplex $D(a)$ defined above. |
212 To each generator $(b, \ol{K})$ of $G_*$ we associate the acyclic subcomplex $D(b)$ defined above. |
215 Both the identity map and $\phi\circ\psi$ are compatible with this |
213 Both the identity map and $\phi\circ\psi$ are compatible with this |
216 collection of acyclic subcomplexes, so by the usual MoAM argument these two maps |
214 collection of acyclic subcomplexes, so by the usual MoAM argument these two maps |
217 are homotopic. |
215 are homotopic. |
218 |
216 |
219 This concludes the proof of Theorem \ref{product_thm}. |
217 This concludes the proof of Theorem \ref{product_thm}. |