3 \section{Action of \texorpdfstring{$\CD{X}$}{$C_*(Diff(M))$}} |
3 \section{Action of \texorpdfstring{$\CD{X}$}{$C_*(Diff(M))$}} |
4 \label{sec:evaluation} |
4 \label{sec:evaluation} |
5 |
5 |
6 Let $CD_*(X, Y)$ denote $C_*(\Diff(X \to Y))$, the singular chain complex of |
6 Let $CD_*(X, Y)$ denote $C_*(\Diff(X \to Y))$, the singular chain complex of |
7 the space of diffeomorphisms |
7 the space of diffeomorphisms |
8 \nn{or homeomorphisms} |
8 \nn{or homeomorphisms; need to fix the diff vs homeo inconsistency} |
9 between the $n$-manifolds $X$ and $Y$ (extending a fixed diffeomorphism $\bd X \to \bd Y$). |
9 between the $n$-manifolds $X$ and $Y$ (extending a fixed diffeomorphism $\bd X \to \bd Y$). |
10 For convenience, we will permit the singular cells generating $CD_*(X, Y)$ to be more general |
10 For convenience, we will permit the singular cells generating $CD_*(X, Y)$ to be more general |
11 than simplices --- they can be based on any linear polyhedron. |
11 than simplices --- they can be based on any linear polyhedron. |
12 \nn{be more restrictive here? does more need to be said?} |
12 \nn{be more restrictive here? does more need to be said?} |
13 We also will use the abbreviated notation $CD_*(X) \deq CD_*(X, X)$. |
13 We also will use the abbreviated notation $CD_*(X) \deq CD_*(X, X)$. |
17 \eq{ |
17 \eq{ |
18 e_{XY} : CD_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) . |
18 e_{XY} : CD_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) . |
19 } |
19 } |
20 On $CD_0(X, Y) \otimes \bc_*(X)$ it agrees with the obvious action of $\Diff(X, Y)$ on $\bc_*(X)$ |
20 On $CD_0(X, Y) \otimes \bc_*(X)$ it agrees with the obvious action of $\Diff(X, Y)$ on $\bc_*(X)$ |
21 (Proposition (\ref{diff0prop})). |
21 (Proposition (\ref{diff0prop})). |
22 For any splittings $X = X_1 \cup X_2$ and $Y = Y_1 \cup Y_2$, |
22 For any compatible splittings $X\to X\sgl$ and $Y\to Y\sgl$, |
23 the following diagram commutes up to homotopy |
23 the following diagram commutes up to homotopy |
24 \eq{ \xymatrix{ |
24 \eq{ \xymatrix{ |
25 CD_*(X, Y) \otimes \bc_*(X) \ar[r]^{e_{XY}} & \bc_*(Y) \\ |
25 CD_*(X\sgl, Y\sgl) \otimes \bc_*(X\sgl) \ar[r]^(.7){e_{X\sgl Y\sgl}} & \bc_*(Y\sgl) \\ |
26 CD_*(X_1, Y_1) \otimes CD_*(X_2, Y_2) \otimes \bc_*(X_1) \otimes \bc_*(X_2) |
26 CD_*(X, Y) \otimes \bc_*(X) |
27 \ar@/_4ex/[r]_{e_{X_1Y_1} \otimes e_{X_2Y_2}} \ar[u]^{\gl \otimes \gl} & |
27 \ar@/_4ex/[r]_{e_{XY}} \ar[u]^{\gl \otimes \gl} & |
28 \bc_*(Y_1) \otimes \bc_*(Y_2) \ar[u]_{\gl} |
28 \bc_*(Y) \ar[u]_{\gl} |
29 } } |
29 } } |
|
30 %For any splittings $X = X_1 \cup X_2$ and $Y = Y_1 \cup Y_2$, |
|
31 %the following diagram commutes up to homotopy |
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32 %\eq{ \xymatrix{ |
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33 % CD_*(X, Y) \otimes \bc_*(X) \ar[r]^{e_{XY}} & \bc_*(Y) \\ |
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34 % CD_*(X_1, Y_1) \otimes CD_*(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 %} } |
30 Any other map satisfying the above two properties is homotopic to $e_X$. |
38 Any other map satisfying the above two properties is homotopic to $e_X$. |
31 \end{prop} |
39 \end{prop} |
32 |
40 |
33 \nn{need to rewrite for self-gluing instead of gluing two pieces together} |
41 \nn{need to rewrite for self-gluing instead of gluing two pieces together} |
34 |
42 |