3 \section{Action of \texorpdfstring{$\CH{X}$}{$C_*(Homeo(M))$}} |
3 \section{Action of \texorpdfstring{$\CH{X}$}{$C_*(Homeo(M))$}} |
4 \label{sec:evaluation} |
4 \label{sec:evaluation} |
5 |
5 |
6 Let $CH_*(X, Y)$ denote $C_*(\Homeo(X \to Y))$, the singular chain complex of |
6 Let $CH_*(X, Y)$ denote $C_*(\Homeo(X \to Y))$, the singular chain complex of |
7 the space of homeomorphisms |
7 the space of homeomorphisms |
8 \nn{need to fix the diff vs homeo inconsistency} |
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9 between the $n$-manifolds $X$ and $Y$ (extending a fixed homeomorphism $\bd X \to \bd Y$). |
8 between the $n$-manifolds $X$ and $Y$ (extending a fixed homeomorphism $\bd X \to \bd Y$). |
10 For convenience, we will permit the singular cells generating $CH_*(X, Y)$ to be more general |
9 For convenience, we will permit the singular cells generating $CH_*(X, Y)$ to be more general |
11 than simplices --- they can be based on any linear polyhedron. |
10 than simplices --- they can be based on any linear polyhedron. |
12 \nn{be more restrictive here? does more need to be said?} |
11 \nn{be more restrictive here? does more need to be said?} |
13 We also will use the abbreviated notation $CH_*(X) \deq CH_*(X, X)$. |
12 We also will use the abbreviated notation $CH_*(X) \deq CH_*(X, X)$. |
14 |
13 |
15 \begin{prop} \label{CHprop} |
14 \begin{prop} \label{CHprop} |
16 For $n$-manifolds $X$ and $Y$ there is a chain map |
15 For $n$-manifolds $X$ and $Y$ there is a chain map |
17 \eq{ |
16 \eq{ |
18 e_{XY} : CH_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) . |
17 e_{XY} : CH_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) |
19 } |
18 } |
20 On $CH_0(X, Y) \otimes \bc_*(X)$ it agrees with the obvious action of $\Homeo(X, Y)$ on $\bc_*(X)$ |
19 such that |
21 (Proposition (\ref{diff0prop})). |
20 \begin{enumerate} |
22 For any compatible splittings $X\to X\sgl$ and $Y\to Y\sgl$, |
21 \item on $CH_0(X, Y) \otimes \bc_*(X)$ it agrees with the obvious action of |
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22 $\Homeo(X, Y)$ on $\bc_*(X)$ (Proposition (\ref{diff0prop})), and |
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23 \item for any compatible splittings $X\to X\sgl$ and $Y\to Y\sgl$, |
23 the following diagram commutes up to homotopy |
24 the following diagram commutes up to homotopy |
24 \eq{ \xymatrix{ |
25 \eq{ \xymatrix{ |
25 CH_*(X\sgl, Y\sgl) \otimes \bc_*(X\sgl) \ar[r]^(.7){e_{X\sgl Y\sgl}} & \bc_*(Y\sgl) \\ |
26 CH_*(X\sgl, Y\sgl) \otimes \bc_*(X\sgl) \ar[r]^(.7){e_{X\sgl Y\sgl}} \ar[d]^{\gl \otimes \gl} & \bc_*(Y\sgl) \ar[d]_{\gl} \\ |
26 CH_*(X, Y) \otimes \bc_*(X) |
27 CH_*(X, Y) \otimes \bc_*(X) |
27 \ar@/_4ex/[r]_{e_{XY}} \ar[u]^{\gl \otimes \gl} & |
28 \ar@/_4ex/[r]_{e_{XY}} & |
28 \bc_*(Y) \ar[u]_{\gl} |
29 \bc_*(Y) |
29 } } |
30 } } |
30 %For any splittings $X = X_1 \cup X_2$ and $Y = Y_1 \cup Y_2$, |
31 \end{enumerate} |
31 %the following diagram commutes up to homotopy |
32 Up to (iterated) homotopy, there is a unique family $\{e_{XY}\}$ of chain maps |
32 %\eq{ \xymatrix{ |
33 satisfying the above two conditions. |
33 % CH_*(X, Y) \otimes \bc_*(X) \ar[r]^{e_{XY}} & \bc_*(Y) \\ |
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34 % CH_*(X_1, Y_1) \otimes CH_*(X_2, Y_2) \otimes \bc_*(X_1) \otimes \bc_*(X_2) |
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35 % \ar@/_4ex/[r]_{e_{X_1Y_1} \otimes e_{X_2Y_2}} \ar[u]^{\gl \otimes \gl} & |
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36 % \bc_*(Y_1) \otimes \bc_*(Y_2) \ar[u]_{\gl} |
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37 %} } |
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38 Any other map satisfying the above two properties is homotopic to $e_X$. |
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39 \end{prop} |
34 \end{prop} |
40 |
35 |
41 \nn{need to rewrite for self-gluing instead of gluing two pieces together} |
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42 |
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43 \nn{Should say something stronger about uniqueness. |
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44 Something like: there is |
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45 a contractible subcomplex of the complex of chain maps |
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46 $CH_*(X) \otimes \bc_*(X) \to \bc_*(X)$ (0-cells are the maps, 1-cells are homotopies, etc.), |
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47 and all choices in the construction lie in the 0-cells of this |
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48 contractible subcomplex. |
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49 Or maybe better to say any two choices are homotopic, and |
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50 any two homotopies and second order homotopic, and so on.} |
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51 |
36 |
52 \nn{Also need to say something about associativity. |
37 \nn{Also need to say something about associativity. |
53 Put it in the above prop or make it a separate prop? |
38 Put it in the above prop or make it a separate prop? |
54 I lean toward the latter.} |
39 I lean toward the latter.} |
55 \medskip |
40 \medskip |
56 |
41 |
57 The proof will occupy the remainder of this section. |
42 The proof will occupy the the next several pages. |
58 \nn{unless we put associativity prop at end} |
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59 |
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60 Without loss of generality, we will assume $X = Y$. |
43 Without loss of generality, we will assume $X = Y$. |
61 |
44 |
62 \medskip |
45 \medskip |
63 |
46 |
64 Let $f: P \times X \to X$ be a family of homeomorphisms and $S \sub X$. |
47 Let $f: P \times X \to X$ be a family of homeomorphisms (e.g. a generator of $CH_*(X)$) |
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48 and let $S \sub X$. |
65 We say that {\it $f$ is supported on $S$} if $f(p, x) = f(q, x)$ for all |
49 We say that {\it $f$ is supported on $S$} if $f(p, x) = f(q, x)$ for all |
66 $x \notin S$ and $p, q \in P$. Equivalently, $f$ is supported on $S$ if there is a family of homeomorphisms $f' : P \times S \to S$ and a `background' |
50 $x \notin S$ and $p, q \in P$. Equivalently, $f$ is supported on $S$ if there is a family of homeomorphisms $f' : P \times S \to S$ and a `background' |
67 homeomorphism $f_0 : X \to X$ so that |
51 homeomorphism $f_0 : X \to X$ so that |
68 \begin{align} |
52 \begin{align} |
69 f(p,s) & = f_0(f'(p,s)) \;\;\;\; \mbox{for}\; (p, s) \in P\times S \\ |
53 f(p,s) & = f_0(f'(p,s)) \;\;\;\; \mbox{for}\; (p, s) \in P\times S \\ |