29 \begin{prop} \label{sympowerprop} |
29 \begin{prop} \label{sympowerprop} |
30 $\bc_*(M, k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$. |
30 $\bc_*(M, k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$. |
31 \end{prop} |
31 \end{prop} |
32 |
32 |
33 \begin{proof} |
33 \begin{proof} |
34 %To define the chain maps between the two complexes we will use the following lemma: |
34 We will use acyclic models (\S \ref{sec:moam}). |
35 % |
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36 %\begin{lemma} |
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37 %Let $A_*$ and $B_*$ be chain complexes, and assume $A_*$ is equipped with |
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38 %a basis (e.g.\ blob diagrams or singular simplices). |
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39 %For each basis element $c \in A_*$ assume given a contractible subcomplex $R(c)_* \sub B_*$ |
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40 %such that $R(c')_* \sub R(c)_*$ whenever $c'$ is a basis element which is part of $\bd c$. |
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41 %Then the complex of chain maps (and (iterated) homotopies) $f:A_*\to B_*$ such that |
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42 %$f(c) \in R(c)_*$ for all $c$ is contractible (and in particular non-empty). |
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43 %\end{lemma} |
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44 % |
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45 %\begin{proof} |
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46 %\nn{easy, but should probably write the details eventually} |
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47 %\nn{this is just the standard ``method of acyclic models" set up, so we should just give a reference for that} |
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48 %\end{proof} |
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49 We will use acyclic models \nn{need ref}. |
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50 Our first task: For each blob diagram $b$ define a subcomplex $R(b)_* \sub C_*(\Sigma^\infty(M))$ |
35 Our first task: For each blob diagram $b$ define a subcomplex $R(b)_* \sub C_*(\Sigma^\infty(M))$ |
51 satisfying the conditions of \nn{need ref}. |
36 satisfying the conditions of Theorem \ref{moam-thm}. |
52 If $b$ is a 0-blob diagram, then it is just a $k[t]$ field on $M$, which is a |
37 If $b$ is a 0-blob diagram, then it is just a $k[t]$ field on $M$, which is a |
53 finite unordered collection of points of $M$ with multiplicities, which is |
38 finite unordered collection of points of $M$ with multiplicities, which is |
54 a point in $\Sigma^\infty(M)$. |
39 a point in $\Sigma^\infty(M)$. |
55 Define $R(b)_*$ to be the singular chain complex of this point. |
40 Define $R(b)_*$ to be the singular chain complex of this point. |
56 If $(B, u, r)$ is an $i$-blob diagram, let $D\sub M$ be its support (the union of the blobs). |
41 If $(B, u, r)$ is an $i$-blob diagram, let $D\sub M$ be its support (the union of the blobs). |
61 The field $r$ on $M\setminus D$ can be thought of as a point in $\Sigma^\infty(M\setminus D)$, |
46 The field $r$ on $M\setminus D$ can be thought of as a point in $\Sigma^\infty(M\setminus D)$, |
62 and using this point we can embed $X$ in $\Sigma^\infty(M)$. |
47 and using this point we can embed $X$ in $\Sigma^\infty(M)$. |
63 Define $R(B, u, r)_*$ to be the singular chain complex of $X$, thought of as a |
48 Define $R(B, u, r)_*$ to be the singular chain complex of $X$, thought of as a |
64 subspace of $\Sigma^\infty(M)$. |
49 subspace of $\Sigma^\infty(M)$. |
65 It is easy to see that $R(\cdot)_*$ satisfies the condition on boundaries from |
50 It is easy to see that $R(\cdot)_*$ satisfies the condition on boundaries from |
66 \nn{need ref, or state condition}. |
51 Theorem \ref{moam-thm}. |
67 Thus we have defined (up to homotopy) a map from |
52 Thus we have defined (up to homotopy) a map from |
68 $\bc_*(M^n, k[t])$ to $C_*(\Sigma^\infty(M))$. |
53 $\bc_*(M, k[t])$ to $C_*(\Sigma^\infty(M))$. |
69 |
54 |
70 Next we define, for each simplex $c$ of $C_*(\Sigma^\infty(M))$, a contractible subspace |
55 Next we define, for each simplex $c$ of $C_*(\Sigma^\infty(M))$, a contractible subspace |
71 $R(c)_* \sub \bc_*(M^n, k[t])$. |
56 $R(c)_* \sub \bc_*(M, k[t])$. |
72 If $c$ is a 0-simplex we use the identification of the fields $\cC(M)$ and |
57 If $c$ is a 0-simplex we use the identification of the fields $\cC(M)$ and |
73 $\Sigma^\infty(M)$ described above. |
58 $\Sigma^\infty(M)$ described above. |
74 Now let $c$ be an $i$-simplex of $\Sigma^j(M)$. |
59 Now let $c$ be an $i$-simplex of $\Sigma^j(M)$. |
75 Choose a metric on $M$, which induces a metric on $\Sigma^j(M)$. |
60 Choose a metric on $M$, which induces a metric on $\Sigma^j(M)$. |
76 We may assume that the diameter of $c$ is small --- that is, $C_*(\Sigma^j(M))$ |
61 We may assume that the diameter of $c$ is small --- that is, $C_*(\Sigma^j(M))$ |
78 How small? $(2r)/3j$, where $r$ is the radius of injectivity of the metric. |
63 How small? $(2r)/3j$, where $r$ is the radius of injectivity of the metric. |
79 Let $T\sub M$ be the ``track" of $c$ in $M$. |
64 Let $T\sub M$ be the ``track" of $c$ in $M$. |
80 \nn{do we need to define this precisely?} |
65 \nn{do we need to define this precisely?} |
81 Choose a neighborhood $D$ of $T$ which is a disjoint union of balls of small diameter. |
66 Choose a neighborhood $D$ of $T$ which is a disjoint union of balls of small diameter. |
82 \nn{need to say more precisely how small} |
67 \nn{need to say more precisely how small} |
83 Define $R(c)_*$ to be $\bc_*(D, k[t]) \sub \bc_*(M^n, k[t])$. |
68 Define $R(c)_*$ to be $\bc_*(D; k[t]) \sub \bc_*(M; k[t])$. |
84 This is contractible by Proposition \ref{bcontract}. |
69 This is contractible by Proposition \ref{bcontract}. |
85 We can arrange that the boundary/inclusion condition is satisfied if we start with |
70 We can arrange that the boundary/inclusion condition is satisfied if we start with |
86 low-dimensional simplices and work our way up. |
71 low-dimensional simplices and work our way up. |
87 \nn{need to be more precise} |
72 \nn{need to be more precise} |
88 |
73 |