1 %!TEX root = ../blob1.tex |
1 %!TEX root = ../blob1.tex |
2 |
2 |
3 \section{Action of \texorpdfstring{$\CD{X}$}{$C_*(Diff(M))$}} |
3 \section{Action of \texorpdfstring{$\CH{X}$}{$C_*(Homeo(M))$}} |
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 $CH_*(X, Y)$ denote $C_*(\Homeo(X \to Y))$, the singular chain complex of |
7 the space of diffeomorphisms |
7 the space of homeomorphisms |
8 \nn{or homeomorphisms; need to fix the diff vs homeo inconsistency} |
8 \nn{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 homeomorphism $\bd X \to \bd Y$). |
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 $CH_*(X, Y)$ 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 $CH_*(X) \deq CH_*(X, X)$. |
14 |
14 |
15 \begin{prop} \label{CDprop} |
15 \begin{prop} \label{CHprop} |
16 For $n$-manifolds $X$ and $Y$ there is a chain map |
16 For $n$-manifolds $X$ and $Y$ there is a chain map |
17 \eq{ |
17 \eq{ |
18 e_{XY} : CD_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) . |
18 e_{XY} : CH_*(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 $CH_0(X, Y) \otimes \bc_*(X)$ it agrees with the obvious action of $\Homeo(X, Y)$ on $\bc_*(X)$ |
21 (Proposition (\ref{diff0prop})). |
21 (Proposition (\ref{diff0prop})). |
22 For any compatible splittings $X\to X\sgl$ and $Y\to Y\sgl$, |
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\sgl, Y\sgl) \otimes \bc_*(X\sgl) \ar[r]^(.7){e_{X\sgl Y\sgl}} & \bc_*(Y\sgl) \\ |
25 CH_*(X\sgl, Y\sgl) \otimes \bc_*(X\sgl) \ar[r]^(.7){e_{X\sgl Y\sgl}} & \bc_*(Y\sgl) \\ |
26 CD_*(X, Y) \otimes \bc_*(X) |
26 CH_*(X, Y) \otimes \bc_*(X) |
27 \ar@/_4ex/[r]_{e_{XY}} \ar[u]^{\gl \otimes \gl} & |
27 \ar@/_4ex/[r]_{e_{XY}} \ar[u]^{\gl \otimes \gl} & |
28 \bc_*(Y) \ar[u]_{\gl} |
28 \bc_*(Y) \ar[u]_{\gl} |
29 } } |
29 } } |
30 %For any splittings $X = X_1 \cup X_2$ and $Y = Y_1 \cup Y_2$, |
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 |
31 %the following diagram commutes up to homotopy |
32 %\eq{ \xymatrix{ |
32 %\eq{ \xymatrix{ |
33 % CD_*(X, Y) \otimes \bc_*(X) \ar[r]^{e_{XY}} & \bc_*(Y) \\ |
33 % CH_*(X, Y) \otimes \bc_*(X) \ar[r]^{e_{XY}} & \bc_*(Y) \\ |
34 % CD_*(X_1, Y_1) \otimes CD_*(X_2, Y_2) \otimes \bc_*(X_1) \otimes \bc_*(X_2) |
34 % CH_*(X_1, Y_1) \otimes CH_*(X_2, Y_2) \otimes \bc_*(X_1) \otimes \bc_*(X_2) |
35 % \ar@/_4ex/[r]_{e_{X_1Y_1} \otimes e_{X_2Y_2}} \ar[u]^{\gl \otimes \gl} & |
35 % \ar@/_4ex/[r]_{e_{X_1Y_1} \otimes e_{X_2Y_2}} \ar[u]^{\gl \otimes \gl} & |
36 % \bc_*(Y_1) \otimes \bc_*(Y_2) \ar[u]_{\gl} |
36 % \bc_*(Y_1) \otimes \bc_*(Y_2) \ar[u]_{\gl} |
37 %} } |
37 %} } |
38 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 $e_X$. |
39 \end{prop} |
39 \end{prop} |
41 \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} |
42 |
42 |
43 \nn{Should say something stronger about uniqueness. |
43 \nn{Should say something stronger about uniqueness. |
44 Something like: there is |
44 Something like: there is |
45 a contractible subcomplex of the complex of chain maps |
45 a contractible subcomplex of the complex of chain maps |
46 $CD_*(X) \otimes \bc_*(X) \to \bc_*(X)$ (0-cells are the maps, 1-cells are homotopies, etc.), |
46 $CH_*(X) \otimes \bc_*(X) \to \bc_*(X)$ (0-cells are the maps, 1-cells are homotopies, etc.), |
47 and all choices in the construction lie in the 0-cells of this |
47 and all choices in the construction lie in the 0-cells of this |
48 contractible subcomplex. |
48 contractible subcomplex. |
49 Or maybe better to say any two choices are homotopic, and |
49 Or maybe better to say any two choices are homotopic, and |
50 any two homotopies and second order homotopic, and so on.} |
50 any two homotopies and second order homotopic, and so on.} |
51 |
51 |
59 |
59 |
60 Without loss of generality, we will assume $X = Y$. |
60 Without loss of generality, we will assume $X = Y$. |
61 |
61 |
62 \medskip |
62 \medskip |
63 |
63 |
64 Let $f: P \times X \to X$ be a family of diffeomorphisms and $S \sub X$. |
64 Let $f: P \times X \to X$ be a family of homeomorphisms and $S \sub X$. |
65 We say that {\it $f$ is supported on $S$} if $f(p, x) = f(q, x)$ for all |
65 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 diffeomorphisms $f' : P \times S \to S$ and a `background' |
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' |
67 diffeomorphism $f_0 : X \to X$ so that |
67 homeomorphism $f_0 : X \to X$ so that |
68 \begin{align} |
68 \begin{align} |
69 f(p,s) & = f_0(f'(p,s)) \;\;\;\; \mbox{for}\; (p, s) \in P\times S \\ |
69 f(p,s) & = f_0(f'(p,s)) \;\;\;\; \mbox{for}\; (p, s) \in P\times S \\ |
70 \intertext{and} |
70 \intertext{and} |
71 f(p,x) & = f_0(x) \;\;\;\; \mbox{for}\; (p, x) \in {P \times (X \setmin S)}. |
71 f(p,x) & = f_0(x) \;\;\;\; \mbox{for}\; (p, x) \in {P \times (X \setmin S)}. |
72 \end{align} |
72 \end{align} |
73 Note that if $f$ is supported on $S$ then it is also supported on any $R \sup S$. |
73 Note that if $f$ is supported on $S$ then it is also supported on any $R \sup S$. |
74 |
74 |
75 Let $\cU = \{U_\alpha\}$ be an open cover of $X$. |
75 Let $\cU = \{U_\alpha\}$ be an open cover of $X$. |
76 A $k$-parameter family of diffeomorphisms $f: P \times X \to X$ is |
76 A $k$-parameter family of homeomorphisms $f: P \times X \to X$ is |
77 {\it adapted to $\cU$} if there is a factorization |
77 {\it adapted to $\cU$} if there is a factorization |
78 \eq{ |
78 \eq{ |
79 P = P_1 \times \cdots \times P_m |
79 P = P_1 \times \cdots \times P_m |
80 } |
80 } |
81 (for some $m \le k$) |
81 (for some $m \le k$) |
82 and families of diffeomorphisms |
82 and families of homeomorphisms |
83 \eq{ |
83 \eq{ |
84 f_i : P_i \times X \to X |
84 f_i : P_i \times X \to X |
85 } |
85 } |
86 such that |
86 such that |
87 \begin{itemize} |
87 \begin{itemize} |
88 \item each $f_i$ is supported on some connected $V_i \sub X$; |
88 \item each $f_i$ is supported on some connected $V_i \sub X$; |
89 \item the sets $V_i$ are mutually disjoint; |
89 \item the sets $V_i$ are mutually disjoint; |
90 \item each $V_i$ is the union of at most $k_i$ of the $U_\alpha$'s, |
90 \item each $V_i$ is the union of at most $k_i$ of the $U_\alpha$'s, |
91 where $k_i = \dim(P_i)$; and |
91 where $k_i = \dim(P_i)$; and |
92 \item $f(p, \cdot) = g \circ f_1(p_1, \cdot) \circ \cdots \circ f_m(p_m, \cdot)$ |
92 \item $f(p, \cdot) = g \circ f_1(p_1, \cdot) \circ \cdots \circ f_m(p_m, \cdot)$ |
93 for all $p = (p_1, \ldots, p_m)$, for some fixed $g \in \Diff(X)$. |
93 for all $p = (p_1, \ldots, p_m)$, for some fixed $g \in \Homeo(X)$. |
94 \end{itemize} |
94 \end{itemize} |
95 A chain $x \in CD_k(X)$ is (by definition) adapted to $\cU$ if it is the sum |
95 A chain $x \in CH_k(X)$ is (by definition) adapted to $\cU$ if it is the sum |
96 of singular cells, each of which is adapted to $\cU$. |
96 of singular cells, each of which is adapted to $\cU$. |
97 |
97 |
98 (Actually, in this section we will only need families of diffeomorphisms to be |
98 (Actually, in this section we will only need families of homeomorphisms to be |
99 {\it weakly adapted} to $\cU$, meaning that the support of $f$ is contained in the union |
99 {\it weakly adapted} to $\cU$, meaning that the support of $f$ is contained in the union |
100 of at most $k$ of the $U_\alpha$'s.) |
100 of at most $k$ of the $U_\alpha$'s.) |
101 |
101 |
102 \begin{lemma} \label{extension_lemma} |
102 \begin{lemma} \label{extension_lemma} |
103 Let $x \in CD_k(X)$ be a singular chain such that $\bd x$ is adapted to $\cU$. |
103 Let $x \in CH_k(X)$ be a singular chain such that $\bd x$ is adapted to $\cU$. |
104 Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$. |
104 Then $x$ is homotopic (rel boundary) to some $x' \in CH_k(X)$ which is adapted to $\cU$. |
105 Furthermore, one can choose the homotopy so that its support is equal to the support of $x$. |
105 Furthermore, one can choose the homotopy so that its support is equal to the support of $x$. |
106 \end{lemma} |
106 \end{lemma} |
107 |
107 |
108 The proof will be given in Section \ref{sec:localising}. |
108 The proof will be given in Section \ref{sec:localising}. |
109 We will actually prove the following more general result. |
109 We will actually prove the following more general result. |
125 \end{lemma} |
125 \end{lemma} |
126 |
126 |
127 |
127 |
128 \medskip |
128 \medskip |
129 |
129 |
130 Before diving into the details, we outline our strategy for the proof of Proposition \ref{CDprop}. |
130 Before diving into the details, we outline our strategy for the proof of Proposition \ref{CHprop}. |
131 |
131 |
132 %Suppose for the moment that evaluation maps with the advertised properties exist. |
132 %Suppose for the moment that evaluation maps with the advertised properties exist. |
133 Let $p$ be a singular cell in $CD_k(X)$ and $b$ be a blob diagram in $\bc_*(X)$. |
133 Let $p$ be a singular cell in $CH_k(X)$ and $b$ be a blob diagram in $\bc_*(X)$. |
134 Suppose that there exists $V \sub X$ such that |
134 Suppose that there exists $V \sub X$ such that |
135 \begin{enumerate} |
135 \begin{enumerate} |
136 \item $V$ is homeomorphic to a disjoint union of balls, and |
136 \item $V$ is homeomorphic to a disjoint union of balls, and |
137 \item $\supp(p) \cup \supp(b) \sub V$. |
137 \item $\supp(p) \cup \supp(b) \sub V$. |
138 \end{enumerate} |
138 \end{enumerate} |
139 Let $W = X \setmin V$, $W' = p(W)$ and $V' = X\setmin W'$. |
139 Let $W = X \setmin V$, $W' = p(W)$ and $V' = X\setmin W'$. |
140 We then have a factorization |
140 We then have a factorization |
141 \[ |
141 \[ |
142 p = \gl(q, r), |
142 p = \gl(q, r), |
143 \] |
143 \] |
144 where $q \in CD_k(V, V')$ and $r \in CD_0(W, W')$. |
144 where $q \in CH_k(V, V')$ and $r \in CH_0(W, W')$. |
145 We can also factorize $b = \gl(b_V, b_W)$, where $b_V\in \bc_*(V)$ and $b_W\in\bc_0(W)$. |
145 We can also factorize $b = \gl(b_V, b_W)$, where $b_V\in \bc_*(V)$ and $b_W\in\bc_0(W)$. |
146 According to the commutative diagram of the proposition, we must have |
146 According to the commutative diagram of the proposition, we must have |
147 \[ |
147 \[ |
148 e_X(p\otimes b) = e_X(\gl(q\otimes b_V, r\otimes b_W)) = |
148 e_X(p\otimes b) = e_X(\gl(q\otimes b_V, r\otimes b_W)) = |
149 gl(e_{VV'}(q\otimes b_V), e_{WW'}(r\otimes b_W)) . |
149 gl(e_{VV'}(q\otimes b_V), e_{WW'}(r\otimes b_W)) . |
150 \] |
150 \] |
151 Since $r$ is a plain, 0-parameter family of diffeomorphisms, we must have |
151 Since $r$ is a plain, 0-parameter family of homeomorphisms, we must have |
152 \[ |
152 \[ |
153 e_{WW'}(r\otimes b_W) = r(b_W), |
153 e_{WW'}(r\otimes b_W) = r(b_W), |
154 \] |
154 \] |
155 where $r(b_W)$ denotes the obvious action of diffeomorphisms on blob diagrams (in |
155 where $r(b_W)$ denotes the obvious action of homeomorphisms on blob diagrams (in |
156 this case a 0-blob diagram). |
156 this case a 0-blob diagram). |
157 Since $V'$ is a disjoint union of balls, $\bc_*(V')$ is acyclic in degrees $>0$ |
157 Since $V'$ is a disjoint union of balls, $\bc_*(V')$ is acyclic in degrees $>0$ |
158 (by \ref{disjunion} and \ref{bcontract}). |
158 (by \ref{disjunion} and \ref{bcontract}). |
159 Assuming inductively that we have already defined $e_{VV'}(\bd(q\otimes b_V))$, |
159 Assuming inductively that we have already defined $e_{VV'}(\bd(q\otimes b_V))$, |
160 there is, up to (iterated) homotopy, a unique choice for $e_{VV'}(q\otimes b_V)$ |
160 there is, up to (iterated) homotopy, a unique choice for $e_{VV'}(q\otimes b_V)$ |
173 To show existence, we must show that the various choices involved in constructing |
173 To show existence, we must show that the various choices involved in constructing |
174 evaluation maps in this way affect the final answer only by a homotopy. |
174 evaluation maps in this way affect the final answer only by a homotopy. |
175 |
175 |
176 \nn{maybe put a little more into the outline before diving into the details.} |
176 \nn{maybe put a little more into the outline before diving into the details.} |
177 |
177 |
178 \nn{Note: At the moment this section is very inconsistent with respect to PL versus smooth, |
178 \nn{Note: At the moment this section is very inconsistent with respect to PL versus smooth, etc |
179 homeomorphism versus diffeomorphism, etc. |
|
180 We expect that everything is true in the PL category, but at the moment our proof |
179 We expect that everything is true in the PL category, but at the moment our proof |
181 avails itself to smooth techniques. |
180 avails itself to smooth techniques. |
182 Furthermore, it traditional in the literature to speak of $C_*(\Diff(X))$ |
181 Furthermore, it traditional in the literature to speak of $C_*(\Diff(X))$ |
183 rather than $C_*(\Homeo(X))$.} |
182 rather than $C_*(\Homeo(X))$.} |
184 |
183 |
193 (e.g.\ $\ep_i = 2^{-i}$). |
192 (e.g.\ $\ep_i = 2^{-i}$). |
194 Choose another sequence of positive real numbers $\delta_i$ such that $\delta_i/\ep_i$ |
193 Choose another sequence of positive real numbers $\delta_i$ such that $\delta_i/\ep_i$ |
195 converges monotonically to zero (e.g.\ $\delta_i = \ep_i^2$). |
194 converges monotonically to zero (e.g.\ $\delta_i = \ep_i^2$). |
196 Let $\phi_l$ be an increasing sequence of positive numbers |
195 Let $\phi_l$ be an increasing sequence of positive numbers |
197 satisfying the inequalities of Lemma \ref{xx2phi}. |
196 satisfying the inequalities of Lemma \ref{xx2phi}. |
198 Given a generator $p\otimes b$ of $CD_*(X)\otimes \bc_*(X)$ and non-negative integers $i$ and $l$ |
197 Given a generator $p\otimes b$ of $CH_*(X)\otimes \bc_*(X)$ and non-negative integers $i$ and $l$ |
199 define |
198 define |
200 \[ |
199 \[ |
201 N_{i,l}(p\ot b) \deq \Nbd_{l\ep_i}(|b|) \cup \Nbd_{\phi_l\delta_i}(|p|). |
200 N_{i,l}(p\ot b) \deq \Nbd_{l\ep_i}(|b|) \cup \Nbd_{\phi_l\delta_i}(|p|). |
202 \] |
201 \] |
203 In other words, we use the metric to choose nested neighborhoods of $|b|\cup |p|$ (parameterized |
202 In other words, we use the metric to choose nested neighborhoods of $|b|\cup |p|$ (parameterized |
204 by $l$), with $\ep_i$ controlling the size of the buffers around $|b|$ and $\delta_i$ controlling |
203 by $l$), with $\ep_i$ controlling the size of the buffers around $|b|$ and $\delta_i$ controlling |
205 the size of the buffers around $|p|$. |
204 the size of the buffers around $|p|$. |
206 |
205 |
207 Next we define subcomplexes $G_*^{i,m} \sub CD_*(X)\otimes \bc_*(X)$. |
206 Next we define subcomplexes $G_*^{i,m} \sub CH_*(X)\otimes \bc_*(X)$. |
208 Let $p\ot b$ be a generator of $CD_*(X)\otimes \bc_*(X)$ and let $k = \deg(p\ot b) |
207 Let $p\ot b$ be a generator of $CH_*(X)\otimes \bc_*(X)$ and let $k = \deg(p\ot b) |
209 = \deg(p) + \deg(b)$. |
208 = \deg(p) + \deg(b)$. |
210 $p\ot b$ is (by definition) in $G_*^{i,m}$ if either (a) $\deg(p) = 0$ or (b) |
209 $p\ot b$ is (by definition) in $G_*^{i,m}$ if either (a) $\deg(p) = 0$ or (b) |
211 there exist codimension-zero submanifolds $V_0,\ldots,V_m \sub X$ such that each $V_j$ |
210 there exist codimension-zero submanifolds $V_0,\ldots,V_m \sub X$ such that each $V_j$ |
212 is homeomorphic to a disjoint union of balls and |
211 is homeomorphic to a disjoint union of balls and |
213 \[ |
212 \[ |
223 $G_*^{i,m}$ is a subcomplex where it is easy to define |
222 $G_*^{i,m}$ is a subcomplex where it is easy to define |
224 the evaluation map. |
223 the evaluation map. |
225 The parameter $m$ controls the number of iterated homotopies we are able to construct |
224 The parameter $m$ controls the number of iterated homotopies we are able to construct |
226 (see Lemma \ref{m_order_hty}). |
225 (see Lemma \ref{m_order_hty}). |
227 The larger $i$ is (i.e.\ the smaller $\ep_i$ is), the better $G_*^{i,m}$ approximates all of |
226 The larger $i$ is (i.e.\ the smaller $\ep_i$ is), the better $G_*^{i,m}$ approximates all of |
228 $CD_*(X)\ot \bc_*(X)$ (see Lemma \ref{Gim_approx}). |
227 $CH_*(X)\ot \bc_*(X)$ (see Lemma \ref{Gim_approx}). |
229 |
228 |
230 Next we define a chain map (dependent on some choices) $e: G_*^{i,m} \to \bc_*(X)$. |
229 Next we define a chain map (dependent on some choices) $e: G_*^{i,m} \to \bc_*(X)$. |
231 Let $p\ot b \in G_*^{i,m}$. |
230 Let $p\ot b \in G_*^{i,m}$. |
232 If $\deg(p) = 0$, define |
231 If $\deg(p) = 0$, define |
233 \[ |
232 \[ |
234 e(p\ot b) = p(b) , |
233 e(p\ot b) = p(b) , |
235 \] |
234 \] |
236 where $p(b)$ denotes the obvious action of the diffeomorphism(s) $p$ on the blob diagram $b$. |
235 where $p(b)$ denotes the obvious action of the homeomorphism(s) $p$ on the blob diagram $b$. |
237 For general $p\ot b$ ($\deg(p) \ge 1$) assume inductively that we have already defined |
236 For general $p\ot b$ ($\deg(p) \ge 1$) assume inductively that we have already defined |
238 $e(p'\ot b')$ when $\deg(p') + \deg(b') < k = \deg(p) + \deg(b)$. |
237 $e(p'\ot b')$ when $\deg(p') + \deg(b') < k = \deg(p) + \deg(b)$. |
239 Choose $V = V_0$ as above so that |
238 Choose $V = V_0$ as above so that |
240 \[ |
239 \[ |
241 N_{i,k}(p\ot b) \subeq V \subeq N_{i,k+1}(p\ot b) . |
240 N_{i,k}(p\ot b) \subeq V \subeq N_{i,k+1}(p\ot b) . |
249 (The second inclusion uses the facts that $|p_j| \subeq |p|$ and $|b_j| \subeq |b|$.) |
248 (The second inclusion uses the facts that $|p_j| \subeq |p|$ and $|b_j| \subeq |b|$.) |
250 We therefore have splittings |
249 We therefore have splittings |
251 \[ |
250 \[ |
252 p = p'\bullet p'' , \;\; b = b'\bullet b'' , \;\; e(\bd(p\ot b)) = f'\bullet b'' , |
251 p = p'\bullet p'' , \;\; b = b'\bullet b'' , \;\; e(\bd(p\ot b)) = f'\bullet b'' , |
253 \] |
252 \] |
254 where $p' \in CD_*(V)$, $p'' \in CD_*(X\setmin V)$, |
253 where $p' \in CH_*(V)$, $p'' \in CH_*(X\setmin V)$, |
255 $b' \in \bc_*(V)$, $b'' \in \bc_*(X\setmin V)$, |
254 $b' \in \bc_*(V)$, $b'' \in \bc_*(X\setmin V)$, |
256 $f' \in \bc_*(p(V))$, and $f'' \in \bc_*(p(X\setmin V))$. |
255 $f' \in \bc_*(p(V))$, and $f'' \in \bc_*(p(X\setmin V))$. |
257 (Note that since the family of diffeomorphisms $p$ is constant (independent of parameters) |
256 (Note that since the family of homeomorphisms $p$ is constant (independent of parameters) |
258 near $\bd V$, the expressions $p(V) \sub X$ and $p(X\setmin V) \sub X$ are |
257 near $\bd V$, the expressions $p(V) \sub X$ and $p(X\setmin V) \sub X$ are |
259 unambiguous.) |
258 unambiguous.) |
260 We have $\deg(p'') = 0$ and, inductively, $f'' = p''(b'')$. |
259 We have $\deg(p'') = 0$ and, inductively, $f'' = p''(b'')$. |
261 %We also have that $\deg(b'') = 0 = \deg(p'')$. |
260 %We also have that $\deg(b'') = 0 = \deg(p'')$. |
262 Choose $x' \in \bc_*(p(V))$ such that $\bd x' = f'$. |
261 Choose $x' \in \bc_*(p(V))$ such that $\bd x' = f'$. |
314 There are splittings |
313 There are splittings |
315 \[ |
314 \[ |
316 p = p'_1\bullet p''_1 , \;\; b = b'_1\bullet b''_1 , |
315 p = p'_1\bullet p''_1 , \;\; b = b'_1\bullet b''_1 , |
317 \;\; e(p\ot b) - \tilde{e}(p\ot b) - h(\bd(p\ot b)) = f'_1\bullet f''_1 , |
316 \;\; e(p\ot b) - \tilde{e}(p\ot b) - h(\bd(p\ot b)) = f'_1\bullet f''_1 , |
318 \] |
317 \] |
319 where $p'_1 \in CD_*(V_1)$, $p''_1 \in CD_*(X\setmin V_1)$, |
318 where $p'_1 \in CH_*(V_1)$, $p''_1 \in CH_*(X\setmin V_1)$, |
320 $b'_1 \in \bc_*(V_1)$, $b''_1 \in \bc_*(X\setmin V_1)$, |
319 $b'_1 \in \bc_*(V_1)$, $b''_1 \in \bc_*(X\setmin V_1)$, |
321 $f'_1 \in \bc_*(p(V_1))$, and $f''_1 \in \bc_*(p(X\setmin V_1))$. |
320 $f'_1 \in \bc_*(p(V_1))$, and $f''_1 \in \bc_*(p(X\setmin V_1))$. |
322 Inductively, $\bd f'_1 = 0$ and $f_1'' = p_1''(b_1'')$. |
321 Inductively, $\bd f'_1 = 0$ and $f_1'' = p_1''(b_1'')$. |
323 Choose $x'_1 \in \bc_*(p(V_1))$ so that $\bd x'_1 = f'_1$. |
322 Choose $x'_1 \in \bc_*(p(V_1))$ so that $\bd x'_1 = f'_1$. |
324 Define |
323 Define |
333 Note that on $G_*^{i,m+1} \subeq G_*^{i,m}$, we have defined two maps, |
332 Note that on $G_*^{i,m+1} \subeq G_*^{i,m}$, we have defined two maps, |
334 call them $e_{i,m}$ and $e_{i,m+1}$. |
333 call them $e_{i,m}$ and $e_{i,m+1}$. |
335 An easy variation on the above lemma shows that $e_{i,m}$ and $e_{i,m+1}$ are $m$-th |
334 An easy variation on the above lemma shows that $e_{i,m}$ and $e_{i,m+1}$ are $m$-th |
336 order homotopic. |
335 order homotopic. |
337 |
336 |
338 Next we show how to homotope chains in $CD_*(X)\ot \bc_*(X)$ to one of the |
337 Next we show how to homotope chains in $CH_*(X)\ot \bc_*(X)$ to one of the |
339 $G_*^{i,m}$. |
338 $G_*^{i,m}$. |
340 Choose a monotone decreasing sequence of real numbers $\gamma_j$ converging to zero. |
339 Choose a monotone decreasing sequence of real numbers $\gamma_j$ converging to zero. |
341 Let $\cU_j$ denote the open cover of $X$ by balls of radius $\gamma_j$. |
340 Let $\cU_j$ denote the open cover of $X$ by balls of radius $\gamma_j$. |
342 Let $h_j: CD_*(X)\to CD_*(X)$ be a chain map homotopic to the identity whose image is spanned by diffeomorphisms with support compatible with $\cU_j$, as described in Lemma \ref{xxxxx}. |
341 Let $h_j: CH_*(X)\to CH_*(X)$ be a chain map homotopic to the identity whose image is spanned by homeomorphisms with support compatible with $\cU_j$, as described in Lemma \ref{xxxxx}. |
343 Recall that $h_j$ and also the homotopy connecting it to the identity do not increase |
342 Recall that $h_j$ and also the homotopy connecting it to the identity do not increase |
344 supports. |
343 supports. |
345 Define |
344 Define |
346 \[ |
345 \[ |
347 g_j \deq h_j\circ h_{j-1} \circ \cdots \circ h_1 . |
346 g_j \deq h_j\circ h_{j-1} \circ \cdots \circ h_1 . |
350 $g_j(p)\ot b$ lies in $G_*^{i,m}$, for arbitrary $m$ and sufficiently large $i$ |
349 $g_j(p)\ot b$ lies in $G_*^{i,m}$, for arbitrary $m$ and sufficiently large $i$ |
351 (depending on $b$, $n = \deg(p)$ and $m$). |
350 (depending on $b$, $n = \deg(p)$ and $m$). |
352 (Note: Don't confuse this $n$ with the top dimension $n$ used elsewhere in this paper.) |
351 (Note: Don't confuse this $n$ with the top dimension $n$ used elsewhere in this paper.) |
353 |
352 |
354 \begin{lemma} \label{Gim_approx} |
353 \begin{lemma} \label{Gim_approx} |
355 Fix a blob diagram $b$, a homotopy order $m$ and a degree $n$ for $CD_*(X)$. |
354 Fix a blob diagram $b$, a homotopy order $m$ and a degree $n$ for $CH_*(X)$. |
356 Then there exists a constant $k_{bmn}$ such that for all $i \ge k_{bmn}$ |
355 Then there exists a constant $k_{bmn}$ such that for all $i \ge k_{bmn}$ |
357 there exists another constant $j_i$ such that for all $j \ge j_i$ and all $p\in CD_n(X)$ |
356 there exists another constant $j_i$ such that for all $j \ge j_i$ and all $p\in CH_n(X)$ |
358 we have $g_j(p)\ot b \in G_*^{i,m}$. |
357 we have $g_j(p)\ot b \in G_*^{i,m}$. |
359 \end{lemma} |
358 \end{lemma} |
360 |
359 |
361 \begin{proof} |
360 \begin{proof} |
362 Let $c$ be a subset of the blobs of $b$. |
361 Let $c$ be a subset of the blobs of $b$. |
385 \[ |
384 \[ |
386 \gamma_j < \delta_i |
385 \gamma_j < \delta_i |
387 \] |
386 \] |
388 and also so that $\phi_t \gamma_j$ is less than the constant $\rho(M)$ of Lemma \ref{xxzz11}. |
387 and also so that $\phi_t \gamma_j$ is less than the constant $\rho(M)$ of Lemma \ref{xxzz11}. |
389 |
388 |
390 Let $j \ge j_i$ and $p\in CD_n(X)$. |
389 Let $j \ge j_i$ and $p\in CH_n(X)$. |
391 Let $q$ be a generator appearing in $g_j(p)$. |
390 Let $q$ be a generator appearing in $g_j(p)$. |
392 Note that $|q|$ is contained in a union of $n$ elements of the cover $\cU_j$, |
391 Note that $|q|$ is contained in a union of $n$ elements of the cover $\cU_j$, |
393 which implies that $|q|$ is contained in a union of $n$ metric balls of radius $\delta_i$. |
392 which implies that $|q|$ is contained in a union of $n$ metric balls of radius $\delta_i$. |
394 We must show that $q\ot b \in G_*^{i,m}$, which means finding neighborhoods |
393 We must show that $q\ot b \in G_*^{i,m}$, which means finding neighborhoods |
395 $V_0,\ldots,V_m \sub X$ of $|q|\cup |b|$ such that each $V_j$ |
394 $V_0,\ldots,V_m \sub X$ of $|q|\cup |b|$ such that each $V_j$ |
514 |
513 |
515 \nn{outline of what remains to be done:} |
514 \nn{outline of what remains to be done:} |
516 |
515 |
517 \begin{itemize} |
516 \begin{itemize} |
518 \item We need to assemble the maps for the various $G^{i,m}$ into |
517 \item We need to assemble the maps for the various $G^{i,m}$ into |
519 a map for all of $CD_*\ot \bc_*$. |
518 a map for all of $CH_*\ot \bc_*$. |
520 One idea: Think of the $g_j$ as a sort of homotopy (from $CD_*\ot \bc_*$ to itself) |
519 One idea: Think of the $g_j$ as a sort of homotopy (from $CH_*\ot \bc_*$ to itself) |
521 parameterized by $[0,\infty)$. For each $p\ot b$ in $CD_*\ot \bc_*$ choose a sufficiently |
520 parameterized by $[0,\infty)$. For each $p\ot b$ in $CH_*\ot \bc_*$ choose a sufficiently |
522 large $j'$. Use these choices to reparameterize $g_\bullet$ so that each |
521 large $j'$. Use these choices to reparameterize $g_\bullet$ so that each |
523 $p\ot b$ gets pushed as far as the corresponding $j'$. |
522 $p\ot b$ gets pushed as far as the corresponding $j'$. |
524 \item Independence of metric, $\ep_i$, $\delta_i$: |
523 \item Independence of metric, $\ep_i$, $\delta_i$: |
525 For a different metric etc. let $\hat{G}^{i,m}$ denote the alternate subcomplexes |
524 For a different metric etc. let $\hat{G}^{i,m}$ denote the alternate subcomplexes |
526 and $\hat{N}_{i,l}$ the alternate neighborhoods. |
525 and $\hat{N}_{i,l}$ the alternate neighborhoods. |