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 |
7 |
8 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 |
9 the space of homeomorphisms |
9 the space of homeomorphisms |
10 between the $n$-manifolds $X$ and $Y$ (extending a fixed homeomorphism $\bd X \to \bd Y$). |
10 between the $n$-manifolds $X$ and $Y$ (any given singular chain extends a fixed homeomorphism $\bd X \to \bd Y$). |
11 We also will use the abbreviated notation $CH_*(X) \deq CH_*(X, X)$. |
11 We also will use the abbreviated notation $CH_*(X) \deq CH_*(X, X)$. |
12 (For convenience, we will permit the singular cells generating $CH_*(X, Y)$ to be more general |
12 (For convenience, we will permit the singular cells generating $CH_*(X, Y)$ to be more general |
13 than simplices --- they can be based on any linear polyhedron. |
13 than simplices --- they can be based on any linear polyhedron. |
14 \nn{be more restrictive here? does more need to be said?}) |
14 \nn{be more restrictive here? does more need to be said?}) |
15 |
15 |
22 \begin{enumerate} |
22 \begin{enumerate} |
23 \item on $CH_0(X, Y) \otimes \bc_*(X)$ it agrees with the obvious action of |
23 \item on $CH_0(X, Y) \otimes \bc_*(X)$ it agrees with the obvious action of |
24 $\Homeo(X, Y)$ on $\bc_*(X)$ (Property (\ref{property:functoriality})), and |
24 $\Homeo(X, Y)$ on $\bc_*(X)$ (Property (\ref{property:functoriality})), and |
25 \item for any compatible splittings $X\to X\sgl$ and $Y\to Y\sgl$, |
25 \item for any compatible splittings $X\to X\sgl$ and $Y\to Y\sgl$, |
26 the following diagram commutes up to homotopy |
26 the following diagram commutes up to homotopy |
27 \eq{ \xymatrix{ |
27 \begin{equation*} |
28 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} \\ |
28 \xymatrix@C+2cm{ |
29 CH_*(X, Y) \otimes \bc_*(X) |
29 CH_*(X, Y) \otimes \bc_*(X) |
30 \ar@/_4ex/[r]_{e_{XY}} & |
30 \ar[r]_(.6){e_{XY}} \ar[d]^{\gl \otimes \gl} & |
31 \bc_*(Y) |
31 \bc_*(Y)\ar[d]^{\gl} \\ |
32 } } |
32 CH_*(X\sgl, Y\sgl) \otimes \bc_*(X\sgl) \ar[r]_(.6){e_{X\sgl Y\sgl}} & \bc_*(Y\sgl) |
|
33 } |
|
34 \end{equation*} |
33 \end{enumerate} |
35 \end{enumerate} |
34 Up to (iterated) homotopy, there is a unique family $\{e_{XY}\}$ of chain maps |
36 Up to (iterated) homotopy, there is a unique family $\{e_{XY}\}$ of chain maps |
35 satisfying the above two conditions. |
37 satisfying the above two conditions. |
36 \end{prop} |
38 \end{prop} |
37 |
39 |
72 The proof will be given in \S\ref{sec:localising}. |
74 The proof will be given in \S\ref{sec:localising}. |
73 |
75 |
74 \medskip |
76 \medskip |
75 |
77 |
76 Before diving into the details, we outline our strategy for the proof of Proposition \ref{CHprop}. |
78 Before diving into the details, we outline our strategy for the proof of Proposition \ref{CHprop}. |
77 |
|
78 %Suppose for the moment that evaluation maps with the advertised properties exist. |
|
79 Let $p$ be a singular cell in $CH_k(X)$ and $b$ be a blob diagram in $\bc_*(X)$. |
79 Let $p$ be a singular cell in $CH_k(X)$ and $b$ be a blob diagram in $\bc_*(X)$. |
80 We say that $p\ot b$ is {\it localizable} if there exists $V \sub X$ such that |
80 We say that $p\ot b$ is {\it localizable} if there exists $V \sub X$ such that |
81 \begin{itemize} |
81 \begin{itemize} |
82 \item $V$ is homeomorphic to a disjoint union of balls, and |
82 \item $V$ is homeomorphic to a disjoint union of balls, and |
83 \item $\supp(p) \cup \supp(b) \sub V$. |
83 \item $\supp(p) \cup \supp(b) \sub V$. |
95 According to the commutative diagram of the proposition, we must have |
95 According to the commutative diagram of the proposition, we must have |
96 \[ |
96 \[ |
97 e_X(p\otimes b) = e_X(\gl(q\otimes b_V, r\otimes b_W)) = |
97 e_X(p\otimes b) = e_X(\gl(q\otimes b_V, r\otimes b_W)) = |
98 gl(e_{VV'}(q\otimes b_V), e_{WW'}(r\otimes b_W)) . |
98 gl(e_{VV'}(q\otimes b_V), e_{WW'}(r\otimes b_W)) . |
99 \] |
99 \] |
100 Since $r$ is a plain, 0-parameter family of homeomorphisms, we must have |
100 Since $r$ is a 0-parameter family of homeomorphisms, we must have |
101 \[ |
101 \[ |
102 e_{WW'}(r\otimes b_W) = r(b_W), |
102 e_{WW'}(r\otimes b_W) = r(b_W), |
103 \] |
103 \] |
104 where $r(b_W)$ denotes the obvious action of homeomorphisms on blob diagrams (in |
104 where $r(b_W)$ denotes the obvious action of homeomorphisms on blob diagrams (in |
105 this case a 0-blob diagram). |
105 this case a 0-blob diagram). |
120 To show existence, we must show that the various choices involved in constructing |
120 To show existence, we must show that the various choices involved in constructing |
121 evaluation maps in this way affect the final answer only by a homotopy. |
121 evaluation maps in this way affect the final answer only by a homotopy. |
122 |
122 |
123 Now for a little more detail. |
123 Now for a little more detail. |
124 (But we're still just motivating the full, gory details, which will follow.) |
124 (But we're still just motivating the full, gory details, which will follow.) |
125 Choose a metric on $X$, and let $\cU_\gamma$ be the open cover of by balls of radius $\gamma$. |
125 Choose a metric on $X$, and let $\cU_\gamma$ be the open cover of balls of radius $\gamma$. |
126 By Lemma \ref{extension_lemma} we can restrict our attention to $k$-parameter families |
126 By Lemma \ref{extension_lemma} we can restrict our attention to $k$-parameter families |
127 $p$ of homeomorphisms such that $\supp(p)$ is contained in the union of $k$ $\gamma$-balls. |
127 $p$ of homeomorphisms such that $\supp(p)$ is contained in the union of $k$ $\gamma$-balls. |
128 For fixed blob diagram $b$ and fixed $k$, it's not hard to show that for $\gamma$ small enough |
128 For fixed blob diagram $b$ and fixed $k$, it's not hard to show that for $\gamma$ small enough |
129 $p\ot b$ must be localizable. |
129 $p\ot b$ must be localizable. |
130 On the other hand, for fixed $k$ and $\gamma$ there exist $p$ and $b$ such that $p\ot b$ is not localizable, |
130 On the other hand, for fixed $k$ and $\gamma$ there exist $p$ and $b$ such that $p\ot b$ is not localizable, |
133 |
133 |
134 The construction of $e_X$, as outlined above, depends on various choices, one of which |
134 The construction of $e_X$, as outlined above, depends on various choices, one of which |
135 is the choice, for each localizable generator $p\ot b$, |
135 is the choice, for each localizable generator $p\ot b$, |
136 of disjoint balls $V$ containing $\supp(p)\cup\supp(b)$. |
136 of disjoint balls $V$ containing $\supp(p)\cup\supp(b)$. |
137 Let $V'$ be another disjoint union of balls containing $\supp(p)\cup\supp(b)$, |
137 Let $V'$ be another disjoint union of balls containing $\supp(p)\cup\supp(b)$, |
138 and assume that there exists yet another disjoint union of balls $W$ with $W$ containing |
138 and assume that there exists yet another disjoint union of balls $W$ containing |
139 $V\cup V'$. |
139 $V\cup V'$. |
140 Then we can use $W$ to construct a homotopy between the two versions of $e_X$ |
140 Then we can use $W$ to construct a homotopy between the two versions of $e_X$ |
141 associated to $V$ and $V'$. |
141 associated to $V$ and $V'$. |
142 If we impose no constraints on $V$ and $V'$ then such a $W$ need not exist. |
142 If we impose no constraints on $V$ and $V'$ then such a $W$ need not exist. |
143 Thus we will insist below that $V$ (and $V'$) be contained in small metric neighborhoods |
143 Thus we will insist below that $V$ (and $V'$) be contained in small metric neighborhoods |
148 |
148 |
149 |
149 |
150 \medskip |
150 \medskip |
151 |
151 |
152 \begin{proof}[Proof of Proposition \ref{CHprop}.] |
152 \begin{proof}[Proof of Proposition \ref{CHprop}.] |
153 Notation: Let $|b| = \supp(b)$, $|p| = \supp(p)$. |
153 We'll use the notation $|b| = \supp(b)$ and $|p| = \supp(p)$. |
154 |
154 |
155 Choose a metric on $X$. |
155 Choose a metric on $X$. |
156 Choose a monotone decreasing sequence of positive real numbers $\ep_i$ converging to zero |
156 Choose a monotone decreasing sequence of positive real numbers $\ep_i$ converging monotonically to zero |
157 (e.g.\ $\ep_i = 2^{-i}$). |
157 (e.g.\ $\ep_i = 2^{-i}$). |
158 Choose another sequence of positive real numbers $\delta_i$ such that $\delta_i/\ep_i$ |
158 Choose another sequence of positive real numbers $\delta_i$ such that $\delta_i/\ep_i$ |
159 converges monotonically to zero (e.g.\ $\delta_i = \ep_i^2$). |
159 converges monotonically to zero (e.g.\ $\delta_i = \ep_i^2$). |
160 Let $\phi_l$ be an increasing sequence of positive numbers |
160 Let $\phi_l$ be an increasing sequence of positive numbers |
161 satisfying the inequalities of Lemma \ref{xx2phi}. |
161 satisfying the inequalities of Lemma \ref{xx2phi} below. |
162 Given a generator $p\otimes b$ of $CH_*(X)\otimes \bc_*(X)$ and non-negative integers $i$ and $l$ |
162 Given a generator $p\otimes b$ of $CH_*(X)\otimes \bc_*(X)$ and non-negative integers $i$ and $l$ |
163 define |
163 define |
164 \[ |
164 \[ |
165 N_{i,l}(p\ot b) \deq \Nbd_{l\ep_i}(|b|) \cup \Nbd_{\phi_l\delta_i}(|p|). |
165 N_{i,l}(p\ot b) \deq \Nbd_{l\ep_i}(|b|) \cup \Nbd_{\phi_l\delta_i}(|p|). |
166 \] |
166 \] |
170 the size of the buffers around $|p|$. |
170 the size of the buffers around $|p|$. |
171 |
171 |
172 Next we define subcomplexes $G_*^{i,m} \sub CH_*(X)\otimes \bc_*(X)$. |
172 Next we define subcomplexes $G_*^{i,m} \sub CH_*(X)\otimes \bc_*(X)$. |
173 Let $p\ot b$ be a generator of $CH_*(X)\otimes \bc_*(X)$ and let $k = \deg(p\ot b) |
173 Let $p\ot b$ be a generator of $CH_*(X)\otimes \bc_*(X)$ and let $k = \deg(p\ot b) |
174 = \deg(p) + \deg(b)$. |
174 = \deg(p) + \deg(b)$. |
175 $p\ot b$ is (by definition) in $G_*^{i,m}$ if either (a) $\deg(p) = 0$ or (b) |
175 We say $p\ot b$ is in $G_*^{i,m}$ exactly when either (a) $\deg(p) = 0$ or (b) |
176 there exist codimension-zero submanifolds $V_0,\ldots,V_m \sub X$ such that each $V_j$ |
176 there exist codimension-zero submanifolds $V_0,\ldots,V_m \sub X$ such that each $V_j$ |
177 is homeomorphic to a disjoint union of balls and |
177 is homeomorphic to a disjoint union of balls and |
178 \[ |
178 \[ |
179 N_{i,k}(p\ot b) \subeq V_0 \subeq N_{i,k+1}(p\ot b) |
179 N_{i,k}(p\ot b) \subeq V_0 \subeq N_{i,k+1}(p\ot b) |
180 \subeq V_1 \subeq \cdots \subeq V_m \subeq N_{i,k+m+1}(p\ot b) . |
180 \subeq V_1 \subeq \cdots \subeq V_m \subeq N_{i,k+m+1}(p\ot b) . |
181 \] |
181 \] |
182 Further, we require (inductively) that $\bd(p\ot b) \in G_*^{i,m}$. |
182 and further $\bd(p\ot b) \in G_*^{i,m}$. |
183 We also require that $b$ is splitable (transverse) along the boundary of each $V_l$. |
183 We also require that $b$ is splitable (transverse) along the boundary of each $V_l$. |
184 |
184 |
185 Note that $G_*^{i,m+1} \subeq G_*^{i,m}$. |
185 Note that $G_*^{i,m+1} \subeq G_*^{i,m}$. |
186 |
186 |
187 As sketched above and explained in detail below, |
187 As sketched above and explained in detail below, |
263 \begin{lemma} \label{m_order_hty} |
263 \begin{lemma} \label{m_order_hty} |
264 Let $\tilde{e} : G_*^{i,m} \to \bc_*(X)$ be a chain map constructed like $e$ above, but with |
264 Let $\tilde{e} : G_*^{i,m} \to \bc_*(X)$ be a chain map constructed like $e$ above, but with |
265 different choices of $V$ (and hence also different choices of $x'$) at each step. |
265 different choices of $V$ (and hence also different choices of $x'$) at each step. |
266 If $m \ge 1$ then $e$ and $\tilde{e}$ are homotopic. |
266 If $m \ge 1$ then $e$ and $\tilde{e}$ are homotopic. |
267 If $m \ge 2$ then any two choices of this first-order homotopy are second-order homotopic. |
267 If $m \ge 2$ then any two choices of this first-order homotopy are second-order homotopic. |
268 And so on. |
268 Continuing, $e : G_*^{i,m} \to \bc_*(X)$ is well-defined up to $m$-th order homotopy. |
269 In other words, $e : G_*^{i,m} \to \bc_*(X)$ is well-defined up to $m$-th order homotopy. |
|
270 \end{lemma} |
269 \end{lemma} |
271 |
270 |
272 \begin{proof} |
271 \begin{proof} |
273 We construct $h: G_*^{i,m} \to \bc_*(X)$ such that $\bd h + h\bd = e - \tilde{e}$. |
272 We construct $h: G_*^{i,m} \to \bc_*(X)$ such that $\bd h + h\bd = e - \tilde{e}$. |
274 $e$ and $\tilde{e}$ coincide on bidegrees $(0, j)$, so define $h$ |
273 The chain maps $e$ and $\tilde{e}$ coincide on bidegrees $(0, j)$, so define $h$ |
275 to be zero there. |
274 to be zero there. |
276 Assume inductively that $h$ has been defined for degrees less than $k$. |
275 Assume inductively that $h$ has been defined for degrees less than $k$. |
277 Let $p\ot b$ be a generator of degree $k$. |
276 Let $p\ot b$ be a generator of degree $k$. |
278 Choose $V_1$ as in the definition of $G_*^{i,m}$ so that |
277 Choose $V_1$ as in the definition of $G_*^{i,m}$ so that |
279 \[ |
278 \[ |
342 \] |
341 \] |
343 for all $l\ge m$ and all $q\ot a$ which appear in the boundary of $p\ot b$. |
342 for all $l\ge m$ and all $q\ot a$ which appear in the boundary of $p\ot b$. |
344 |
343 |
345 |
344 |
346 \begin{proof} |
345 \begin{proof} |
347 Let $c$ be a subset of the blobs of $b$. |
346 |
348 There exists $\lambda > 0$ such that $\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < \lambda$ |
347 There exists $\lambda > 0$ such that for every subset $c$ of the blobs of $b$ $\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < \lambda$ . |
349 and all such $c$. |
348 (Here we are using the fact that the blobs are piecewise-linear and thatthat $\bd c$ is collared.) |
350 (Here we are using a piecewise smoothness assumption for $\bd c$, and also |
|
351 the fact that $\bd c$ is collared. |
|
352 We need to consider all such $c$ because all generators appearing in |
349 We need to consider all such $c$ because all generators appearing in |
353 iterated boundaries of $p\ot b$ must be in $G_*^{i,m}$.) |
350 iterated boundaries of $p\ot b$ must be in $G_*^{i,m}$.) |
354 |
351 |
355 Let $r = \deg(b)$ and |
352 Let $r = \deg(b)$ and |
356 \[ |
353 \[ |
412 |
409 |
413 The same argument shows that each generator involved in iterated boundaries of $q\ot b$ |
410 The same argument shows that each generator involved in iterated boundaries of $q\ot b$ |
414 is in $G_*^{i,m}$. |
411 is in $G_*^{i,m}$. |
415 \end{proof} |
412 \end{proof} |
416 |
413 |
417 In the next few lemmas we have made no effort to optimize the various bounds. |
414 In the next three lemmas, which provide the estimates needed above, we have made no effort to optimize the various bounds. |
418 (The bounds are, however, optimal in the sense of minimizing the amount of work |
415 (The bounds are, however, optimal in the sense of minimizing the amount of work |
419 we do. Equivalently, they are the first bounds we thought of.) |
416 we do. Equivalently, they are the first bounds we thought of.) |
420 |
417 |
421 We say that a subset $S$ of a metric space has radius $\le r$ if $S$ is contained in |
418 We say that a subset $S$ of a metric space has radius $\le r$ if $S$ is contained in |
422 some metric ball of radius $r$. |
419 some metric ball of radius $r$. |
429 \begin{proof} \label{xxyy2} |
426 \begin{proof} \label{xxyy2} |
430 Let $S$ be contained in $B_r(y)$, $y \in \ebb^n$. |
427 Let $S$ be contained in $B_r(y)$, $y \in \ebb^n$. |
431 Note that if $a \ge 2r$ then $\Nbd_a(S) \sup B_r(y)$. |
428 Note that if $a \ge 2r$ then $\Nbd_a(S) \sup B_r(y)$. |
432 Let $z\in \Nbd_a(S) \setmin B_r(y)$. |
429 Let $z\in \Nbd_a(S) \setmin B_r(y)$. |
433 Consider the triangle |
430 Consider the triangle |
434 \nn{give figure?} with vertices $z$, $y$ and $s$ with $s\in S$. |
431 with vertices $z$, $y$ and $s$ with $s\in S$. |
435 The length of the edge $yz$ is greater than $r$ which is greater |
432 The length of the edge $yz$ is greater than $r$ which is greater |
436 than the length of the edge $ys$. |
433 than the length of the edge $ys$. |
437 It follows that the angle at $z$ is less than $\pi/2$ (less than $\pi/3$, in fact), |
434 It follows that the angle at $z$ is less than $\pi/2$ (less than $\pi/3$, in fact), |
438 which means that points on the edge $yz$ near $z$ are closer to $s$ than $z$ is, |
435 which means that points on the edge $yz$ near $z$ are closer to $s$ than $z$ is, |
439 which implies that these points are also in $\Nbd_a(S)$. |
436 which implies that these points are also in $\Nbd_a(S)$. |
440 Hence $\Nbd_a(S)$ is star-shaped with respect to $y$. |
437 Hence $\Nbd_a(S)$ is star-shaped with respect to $y$. |
441 \end{proof} |
438 \end{proof} |
442 |
439 |
443 If we replace $\ebb^n$ above with an arbitrary compact Riemannian manifold $M$, |
440 If we replace $\ebb^n$ above with an arbitrary compact Riemannian manifold $M$, |
444 the same result holds, so long as $a$ is not too large: |
441 the same result holds, so long as $a$ is not too large: |
445 \nn{what about PL? TOP?} |
442 \nn{replace this with a PL version} |
446 |
443 |
447 \begin{lemma} \label{xxzz11} |
444 \begin{lemma} \label{xxzz11} |
448 Let $M$ be a compact Riemannian manifold. |
445 Let $M$ be a compact Riemannian manifold. |
449 Then there is a constant $\rho(M)$ such that for all |
446 Then there is a constant $\rho(M)$ such that for all |
450 subsets $S\sub M$ of radius $\le r$ and all $a$ such that $2r \le a \le \rho(M)$, |
447 subsets $S\sub M$ of radius $\le r$ and all $a$ such that $2r \le a \le \rho(M)$, |
496 U \subeq \Nbd_{\phi_{k-1}r'}(R) = \Nbd_{t}(S) \subeq \Nbd_{\phi_k r}(S) , |
493 U \subeq \Nbd_{\phi_{k-1}r'}(R) = \Nbd_{t}(S) \subeq \Nbd_{\phi_k r}(S) , |
497 \] |
494 \] |
498 where $t = \phi_{k-1}(2\phi_{k-1}+2)r + \phi_{k-1} r$. |
495 where $t = \phi_{k-1}(2\phi_{k-1}+2)r + \phi_{k-1} r$. |
499 \end{proof} |
496 \end{proof} |
500 |
497 |
501 \medskip |
498 |
|
499 We now return to defining the chain maps $e_X$. |
|
500 |
502 |
501 |
503 Let $R_*$ be the chain complex with a generating 0-chain for each non-negative |
502 Let $R_*$ be the chain complex with a generating 0-chain for each non-negative |
504 integer and a generating 1-chain connecting each adjacent pair $(j, j+1)$. |
503 integer and a generating 1-chain connecting each adjacent pair $(j, j+1)$. |
505 (So $R_*$ is a simplicial version of the non-negative reals.) |
504 (So $R_*$ is a simplicial version of the non-negative reals.) |
506 Denote the 0-chains by $j$ (for $j$ a non-negative integer) and the 1-chain connecting $j$ and $j+1$ |
505 Denote the 0-chains by $j$ (for $j$ a non-negative integer) and the 1-chain connecting $j$ and $j+1$ |
590 Next we show that the action maps are compatible with gluing. |
589 Next we show that the action maps are compatible with gluing. |
591 Let $G^m_*$ and $\ol{G}^m_*$ be the complexes, as above, used for defining |
590 Let $G^m_*$ and $\ol{G}^m_*$ be the complexes, as above, used for defining |
592 the action maps $e_{X\sgl}$ and $e_X$. |
591 the action maps $e_{X\sgl}$ and $e_X$. |
593 The gluing map $X\sgl\to X$ induces a map |
592 The gluing map $X\sgl\to X$ induces a map |
594 \[ |
593 \[ |
595 \gl: R_*\ot CH_*(X\sgl, X \sgl) \otimes \bc_*(X \sgl) \to R_*\ot CH_*(X, X) \otimes \bc_*(X) , |
594 \gl: R_*\ot CH_*(X, X) \otimes \bc_*(X) \to R_*\ot CH_*(X\sgl, X \sgl) \otimes \bc_*(X \sgl) , |
596 \] |
595 \] |
597 and it is easy to see that $\gl(G^m_*)\sub \ol{G}^m_*$. |
596 and it is easy to see that $\gl(G^m_*)\sub \ol{G}^m_*$. |
598 From this it follows that the diagram in the statement of Proposition \ref{CHprop} commutes. |
597 From this it follows that the diagram in the statement of Proposition \ref{CHprop} commutes. |
599 |
598 |
600 \medskip |
599 \todo{this paragraph isn't very convincing, or at least I don't see what's going on} |
601 |
|
602 Finally we show that the action maps defined above are independent of |
600 Finally we show that the action maps defined above are independent of |
603 the choice of metric (up to iterated homotopy). |
601 the choice of metric (up to iterated homotopy). |
604 The arguments are very similar to ones given above, so we only sketch them. |
602 The arguments are very similar to ones given above, so we only sketch them. |
605 Let $g$ and $g'$ be two metrics on $X$, and let $e$ and $e'$ be the corresponding |
603 Let $g$ and $g'$ be two metrics on $X$, and let $e$ and $e'$ be the corresponding |
606 actions $CH_*(X, X) \ot \bc_*(X)\to\bc_*(X)$. |
604 actions $CH_*(X, X) \ot \bc_*(X)\to\bc_*(X)$. |
612 by $p\ot b$ such that $|p|\cup|b|$ is contained in a disjoint union of balls. |
610 by $p\ot b$ such that $|p|\cup|b|$ is contained in a disjoint union of balls. |
613 Using acyclic models, we can construct a homotopy from $e$ to $e'$ on $F_*$. |
611 Using acyclic models, we can construct a homotopy from $e$ to $e'$ on $F_*$. |
614 We now observe that $CH_*(X, X) \ot \bc_*(X)$ retracts to $F_*$. |
612 We now observe that $CH_*(X, X) \ot \bc_*(X)$ retracts to $F_*$. |
615 Similar arguments show that this homotopy from $e$ to $e'$ is well-defined |
613 Similar arguments show that this homotopy from $e$ to $e'$ is well-defined |
616 up to second order homotopy, and so on. |
614 up to second order homotopy, and so on. |
|
615 |
|
616 This completes the proof of Proposition \ref{CHprop}. |
617 \end{proof} |
617 \end{proof} |
618 |
618 |
619 |
619 |
620 \begin{rem*} |
620 \begin{rem*} |
621 \label{rem:for-small-blobs} |
621 \label{rem:for-small-blobs} |
622 For the proof of Lemma \ref{lem:CH-small-blobs} below we will need the following observation on the action constructed above. |
622 For the proof of Lemma \ref{lem:CH-small-blobs} below we will need the following observation on the action constructed above. |
623 Let $b$ be a blob diagram and $p:P\times X\to X$ be a family of homeomorphisms. |
623 Let $b$ be a blob diagram and $p:P\times X\to X$ be a family of homeomorphisms. |
624 Then we may choose $e$ such that $e(p\ot b)$ is a sum of generators, each |
624 Then we may choose $e$ such that $e(p\ot b)$ is a sum of generators, each |
625 of which has support close to $p(t,|b|)$ for some $t\in P$. |
625 of which has support close to $p(t,|b|)$ for some $t\in P$. |
626 More precisely, the support of the generators is contained in a small neighborhood |
626 More precisely, the support of the generators is contained in the union of a small neighborhood |
627 of $p(t,|b|)$ union some small balls. |
627 of $p(t,|b|)$ with some small balls. |
628 (Here ``small" is in terms of the metric on $X$ that we chose to construct $e$.) |
628 (Here ``small" is in terms of the metric on $X$ that we chose to construct $e$.) |
629 \end{rem*} |
629 \end{rem*} |
630 |
630 |
631 |
631 |
632 \begin{prop} |
632 \begin{prop} |