2 |
2 |
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 \nn{should comment at the start about any assumptions about smooth, PL etc.} |
6 \nn{should comment at the start about any assumptions about smooth, PL etc.} |
7 |
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8 \noop{Note: At the moment this section is very inconsistent with respect to PL versus smooth, etc |
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9 We expect that everything is true in the PL category, but at the moment our proof |
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10 avails itself to smooth techniques. |
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11 Furthermore, it traditional in the literature to speak of $C_*(\Diff(X))$ |
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12 rather than $C_*(\Homeo(X))$.} |
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13 |
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14 |
7 |
15 Let $CH_*(X, Y)$ denote $C_*(\Homeo(X \to Y))$, the singular chain complex of |
8 Let $CH_*(X, Y)$ denote $C_*(\Homeo(X \to Y))$, the singular chain complex of |
16 the space of homeomorphisms |
9 the space of homeomorphisms |
17 between the $n$-manifolds $X$ and $Y$ (extending a fixed homeomorphism $\bd X \to \bd Y$). |
10 between the $n$-manifolds $X$ and $Y$ (extending a fixed homeomorphism $\bd X \to \bd Y$). |
18 For convenience, we will permit the singular cells generating $CH_*(X, Y)$ to be more general |
11 We also will use the abbreviated notation $CH_*(X) \deq CH_*(X, X)$. |
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12 (For convenience, we will permit the singular cells generating $CH_*(X, Y)$ to be more general |
19 than simplices --- they can be based on any linear polyhedron. |
13 than simplices --- they can be based on any linear polyhedron. |
20 \nn{be more restrictive here? does more need to be said?} |
14 \nn{be more restrictive here? does more need to be said?}) |
21 We also will use the abbreviated notation $CH_*(X) \deq CH_*(X, X)$. |
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22 |
15 |
23 \begin{prop} \label{CHprop} |
16 \begin{prop} \label{CHprop} |
24 For $n$-manifolds $X$ and $Y$ there is a chain map |
17 For $n$-manifolds $X$ and $Y$ there is a chain map |
25 \eq{ |
18 \eq{ |
26 e_{XY} : CH_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) |
19 e_{XY} : CH_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) |
202 The parameter $m$ controls the number of iterated homotopies we are able to construct |
195 The parameter $m$ controls the number of iterated homotopies we are able to construct |
203 (see Lemma \ref{m_order_hty}). |
196 (see Lemma \ref{m_order_hty}). |
204 The larger $i$ is (i.e.\ the smaller $\ep_i$ is), the better $G_*^{i,m}$ approximates all of |
197 The larger $i$ is (i.e.\ the smaller $\ep_i$ is), the better $G_*^{i,m}$ approximates all of |
205 $CH_*(X)\ot \bc_*(X)$ (see Lemma \ref{Gim_approx}). |
198 $CH_*(X)\ot \bc_*(X)$ (see Lemma \ref{Gim_approx}). |
206 |
199 |
207 Next we define a chain map (dependent on some choices) $e: G_*^{i,m} \to \bc_*(X)$. |
200 Next we define a chain map (dependent on some choices) $e_{i,m}: G_*^{i,m} \to \bc_*(X)$. |
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201 (When the domain is clear from context we will drop the subscripts and write |
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202 simply $e: G_*^{i,m} \to \bc_*(X)$). |
208 Let $p\ot b \in G_*^{i,m}$. |
203 Let $p\ot b \in G_*^{i,m}$. |
209 If $\deg(p) = 0$, define |
204 If $\deg(p) = 0$, define |
210 \[ |
205 \[ |
211 e(p\ot b) = p(b) , |
206 e(p\ot b) = p(b) , |
212 \] |
207 \] |
307 |
302 |
308 The $j$-th order homotopy is constructed similarly, with $V_j$ replacing $V_1$ above. |
303 The $j$-th order homotopy is constructed similarly, with $V_j$ replacing $V_1$ above. |
309 \end{proof} |
304 \end{proof} |
310 |
305 |
311 Note that on $G_*^{i,m+1} \subeq G_*^{i,m}$, we have defined two maps, |
306 Note that on $G_*^{i,m+1} \subeq G_*^{i,m}$, we have defined two maps, |
312 call them $e_{i,m}$ and $e_{i,m+1}$. |
307 $e_{i,m}$ and $e_{i,m+1}$. |
313 An easy variation on the above lemma shows that $e_{i,m}$ and $e_{i,m+1}$ are $m$-th |
308 An easy variation on the above lemma shows that |
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309 the restrictions of $e_{i,m}$ and $e_{i,m+1}$ to $G_*^{i,m+1}$ are $m$-th |
314 order homotopic. |
310 order homotopic. |
315 |
311 |
316 Next we show how to homotope chains in $CH_*(X)\ot \bc_*(X)$ to one of the |
312 Next we show how to homotope chains in $CH_*(X)\ot \bc_*(X)$ to one of the |
317 $G_*^{i,m}$. |
313 $G_*^{i,m}$. |
318 Choose a monotone decreasing sequence of real numbers $\gamma_j$ converging to zero. |
314 Choose a monotone decreasing sequence of real numbers $\gamma_j$ converging to zero. |
433 Hence $\Nbd_a(S)$ is star-shaped with respect to $y$. |
429 Hence $\Nbd_a(S)$ is star-shaped with respect to $y$. |
434 \end{proof} |
430 \end{proof} |
435 |
431 |
436 If we replace $\ebb^n$ above with an arbitrary compact Riemannian manifold $M$, |
432 If we replace $\ebb^n$ above with an arbitrary compact Riemannian manifold $M$, |
437 the same result holds, so long as $a$ is not too large: |
433 the same result holds, so long as $a$ is not too large: |
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434 \nn{what about PL? TOP?} |
438 |
435 |
439 \begin{lemma} \label{xxzz11} |
436 \begin{lemma} \label{xxzz11} |
440 Let $M$ be a compact Riemannian manifold. |
437 Let $M$ be a compact Riemannian manifold. |
441 Then there is a constant $\rho(M)$ such that for all |
438 Then there is a constant $\rho(M)$ such that for all |
442 subsets $S\sub M$ of radius $\le r$ and all $a$ such that $2r \le a \le \rho(M)$, |
439 subsets $S\sub M$ of radius $\le r$ and all $a$ such that $2r \le a \le \rho(M)$, |