14 |
14 |
15 \medskip |
15 \medskip |
16 |
16 |
17 An important technical tool in the proofs of this section is provided by the idea of ``small blobs". |
17 An important technical tool in the proofs of this section is provided by the idea of ``small blobs". |
18 Fix $\cU$, an open cover of $M$. |
18 Fix $\cU$, an open cover of $M$. |
19 Define the ``small blob complex" $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(M)$ of all blob diagrams in which every blob is contained in some open set of $\cU$, and moreover each field labeling a region cut out by the blobs is splittable into fields on smaller regions, each of which is contained in some open set of $\cU$. |
19 Define the ``small blob complex" $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(M)$ |
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20 of all blob diagrams in which every blob is contained in some open set of $\cU$, |
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21 and moreover each field labeling a region cut out by the blobs is splittable |
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22 into fields on smaller regions, each of which is contained in some open set of $\cU$. |
20 |
23 |
21 \begin{thm}[Small blobs] \label{thm:small-blobs} |
24 \begin{thm}[Small blobs] \label{thm:small-blobs} |
22 The inclusion $i: \bc^{\cU}_*(M) \into \bc_*(M)$ is a homotopy equivalence. |
25 The inclusion $i: \bc^{\cU}_*(M) \into \bc_*(M)$ is a homotopy equivalence. |
23 \end{thm} |
26 \end{thm} |
24 The proof appears in \S \ref{appendix:small-blobs}. |
27 The proof appears in \S \ref{appendix:small-blobs}. |
46 |
49 |
47 First we define a map |
50 First we define a map |
48 \[ |
51 \[ |
49 \psi: \cl{\cC_F}(Y) \to \bc_*(Y\times F;C) . |
52 \psi: \cl{\cC_F}(Y) \to \bc_*(Y\times F;C) . |
50 \] |
53 \] |
51 In filtration degree 0 we just glue together the various blob diagrams on $X_i\times F$ |
54 On 0-simplices of the hocolimit |
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55 we just glue together the various blob diagrams on $X_i\times F$ |
52 (where $X_i$ is a component of a permissible decomposition of $Y$) to get a blob diagram on |
56 (where $X_i$ is a component of a permissible decomposition of $Y$) to get a blob diagram on |
53 $Y\times F$. |
57 $Y\times F$. |
54 In filtration degrees 1 and higher we define the map to be zero. |
58 For simplices of dimension 1 and higher we define the map to be zero. |
55 It is easy to check that this is a chain map. |
59 It is easy to check that this is a chain map. |
56 |
60 |
57 In the other direction, we will define a subcomplex $G_*\sub \bc_*(Y\times F;C)$ |
61 In the other direction, we will define a subcomplex $G_*\sub \bc_*(Y\times F;C)$ |
58 and a map |
62 and a map |
59 \[ |
63 \[ |
78 (Recall that $\ol{K} = (K_0,\ldots,K_l)$ denotes a chain of decompositions; |
82 (Recall that $\ol{K} = (K_0,\ldots,K_l)$ denotes a chain of decompositions; |
79 see \S\ref{ss:ncat_fields}.) |
83 see \S\ref{ss:ncat_fields}.) |
80 By $(b, \ol{K})$ we really mean $(b^\sharp, \ol{K})$, where $b^\sharp$ is |
84 By $(b, \ol{K})$ we really mean $(b^\sharp, \ol{K})$, where $b^\sharp$ is |
81 $b$ split according to $K_0\times F$. |
85 $b$ split according to $K_0\times F$. |
82 To simplify notation we will just write plain $b$ instead of $b^\sharp$. |
86 To simplify notation we will just write plain $b$ instead of $b^\sharp$. |
83 Roughly speaking, $D(a)$ consists of filtration degree 0 stuff which glues up to give |
87 Roughly speaking, $D(a)$ consists of 0-simplices which glue up to give |
84 $a$ (or one of its iterated boundaries), filtration degree 1 stuff which makes all of the filtration degree 0 stuff homologous, |
88 $a$ (or one of its iterated boundaries), 1-simplices which connect all the 0-simplices, |
85 filtration degree 2 stuff which kills the homology created by the |
89 2-simplices which kill the homology created by the |
86 filtration degree 1 stuff, and so on. |
90 1-simplices, and so on. |
87 More formally, |
91 More formally, |
88 |
92 |
89 \begin{lemma} \label{lem:d-a-acyclic} |
93 \begin{lemma} \label{lem:d-a-acyclic} |
90 $D(a)$ is acyclic. |
94 $D(a)$ is acyclic. |
91 \end{lemma} |
95 \end{lemma} |
92 |
96 |
93 \begin{proof} |
97 \begin{proof} |
94 We will prove acyclicity in the first couple of degrees, and \nn{in this draft, at least} |
98 We will prove acyclicity in the first couple of degrees, and \nn{in this draft, at least} |
95 leave the general case to the reader. |
99 leave the general case to the reader. |
96 |
100 |
97 Let $K$ and $K'$ be two decompositions of $Y$ compatible with $a$. |
101 Let $K$ and $K'$ be two decompositions (0-simplices) of $Y$ compatible with $a$. |
98 We want to show that $(a, K)$ and $(a, K')$ are homologous via filtration degree 1 stuff. |
102 We want to find 1-simplices which connect $K$ and $K'$. |
99 \nn{need to say this better; these two chains don't have the same boundary.} |
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100 We might hope that $K$ and $K'$ have a common refinement, but this is not necessarily |
103 We might hope that $K$ and $K'$ have a common refinement, but this is not necessarily |
101 the case. |
104 the case. |
102 (Consider the $x$-axis and the graph of $y = x^2\sin(1/x)$ in $\r^2$.) |
105 (Consider the $x$-axis and the graph of $y = x^2\sin(1/x)$ in $\r^2$.) |
103 However, we {\it can} find another decomposition $L$ such that $L$ shares common |
106 However, we {\it can} find another decomposition $L$ such that $L$ shares common |
104 refinements with both $K$ and $K'$. |
107 refinements with both $K$ and $K'$. |
105 Let $KL$ and $K'L$ denote these two refinements. |
108 Let $KL$ and $K'L$ denote these two refinements. |
106 Then filtration degree 1 chains associated to the four anti-refinements |
109 Then 1-simplices associated to the four anti-refinements |
107 $KL\to K$, $KL\to L$, $K'L\to L$ and $K'L\to K'$ |
110 $KL\to K$, $KL\to L$, $K'L\to L$ and $K'L\to K'$ |
108 give the desired chain connecting $(a, K)$ and $(a, K')$ |
111 give the desired chain connecting $(a, K)$ and $(a, K')$ |
109 (see Figure \ref{zzz4}). |
112 (see Figure \ref{zzz4}). |
110 |
113 |
111 \begin{figure}[!ht] |
114 \begin{figure}[!ht] |
124 \caption{Connecting $K$ and $K'$ via $L$} |
127 \caption{Connecting $K$ and $K'$ via $L$} |
125 \label{zzz4} |
128 \label{zzz4} |
126 \end{figure} |
129 \end{figure} |
127 |
130 |
128 Consider a different choice of decomposition $L'$ in place of $L$ above. |
131 Consider a different choice of decomposition $L'$ in place of $L$ above. |
129 This leads to a cycle consisting of filtration degree 1 stuff. |
132 This leads to a cycle of 1-simplices. |
130 We want to show that this cycle bounds a chain of filtration degree 2 stuff. |
133 We want to find 2-simplices which fill in this cycle. |
131 Choose a decomposition $M$ which has common refinements with each of |
134 Choose a decomposition $M$ which has common refinements with each of |
132 $K$, $KL$, $L$, $K'L$, $K'$, $K'L'$, $L'$ and $KL'$. |
135 $K$, $KL$, $L$, $K'L$, $K'$, $K'L'$, $L'$ and $KL'$. |
133 (We also also require that $KLM$ antirefines to $KM$, etc.) |
136 (We also also require that $KLM$ antirefines to $KM$, etc.) |
134 Then we have a filtration degree 2 chain, as shown in Figure \ref{zzz5}, which does the trick. |
137 Then we have 2-simplices, as shown in Figure \ref{zzz5}, which do the trick. |
135 (Each small triangle in Figure \ref{zzz5} can be filled with a filtration degree 2 chain.) |
138 (Each small triangle in Figure \ref{zzz5} can be filled with a 2-simplex.) |
136 |
139 |
137 \begin{figure}[!ht] |
140 \begin{figure}[!ht] |
138 %\begin{equation*} |
141 %\begin{equation*} |
139 %\mathfig{1.0}{tempkw/zz5} |
142 %\mathfig{1.0}{tempkw/zz5} |
140 %\end{equation*} |
143 %\end{equation*} |
177 Continuing in this way we see that $D(a)$ is acyclic. |
180 Continuing in this way we see that $D(a)$ is acyclic. |
178 \end{proof} |
181 \end{proof} |
179 |
182 |
180 We are now in a position to apply the method of acyclic models to get a map |
183 We are now in a position to apply the method of acyclic models to get a map |
181 $\phi:G_* \to \cl{\cC_F}(Y)$. |
184 $\phi:G_* \to \cl{\cC_F}(Y)$. |
182 We may assume that $\phi(a)$ has the form $(a, K) + r$, where $(a, K)$ is in filtration degree zero |
185 We may assume that $\phi(a)$ has the form $(a, K) + r$, where $(a, K)$ is a 0-simplex |
183 and $r$ has filtration degree greater than zero. |
186 and $r$ is a sum of simplices of dimension 1 or higher. |
184 |
187 |
185 We now show that $\phi\circ\psi$ and $\psi\circ\phi$ are homotopic to the identity. |
188 We now show that $\phi\circ\psi$ and $\psi\circ\phi$ are homotopic to the identity. |
186 |
189 |
187 First, $\psi\circ\phi$ is the identity on the nose: |
190 First, $\psi\circ\phi$ is the identity on the nose: |
188 \[ |
191 \[ |
189 \psi(\phi(a)) = \psi((a,K)) + \psi(r) = a + 0. |
192 \psi(\phi(a)) = \psi((a,K)) + \psi(r) = a + 0. |
190 \] |
193 \] |
191 Roughly speaking, $(a, K)$ is just $a$ chopped up into little pieces, and |
194 Roughly speaking, $(a, K)$ is just $a$ chopped up into little pieces, and |
192 $\psi$ glues those pieces back together, yielding $a$. |
195 $\psi$ glues those pieces back together, yielding $a$. |
193 We have $\psi(r) = 0$ since $\psi$ is zero in positive filtration degrees. |
196 We have $\psi(r) = 0$ since $\psi$ is zero on $(\ge 1)$-simplices. |
194 |
197 |
195 Second, $\phi\circ\psi$ is the identity up to homotopy by another argument based on the method of acyclic models. |
198 Second, $\phi\circ\psi$ is the identity up to homotopy by another argument based on the method of acyclic models. |
196 To each generator $(b, \ol{K})$ of $G_*$ we associate the acyclic subcomplex $D(b)$ defined above. |
199 To each generator $(b, \ol{K})$ of $G_*$ we associate the acyclic subcomplex $D(b)$ defined above. |
197 Both the identity map and $\phi\circ\psi$ are compatible with this |
200 Both the identity map and $\phi\circ\psi$ are compatible with this |
198 collection of acyclic subcomplexes, so by the usual method of acyclic models argument these two maps |
201 collection of acyclic subcomplexes, so by the usual method of acyclic models argument these two maps |