1 %!TEX root = ../blob1.tex |
1 %!TEX root = ../blob1.tex |
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 |
|
6 \nn{should comment at the start about any assumptions about smooth, PL etc.} |
|
7 |
|
8 \noop{Note: At the moment this section is very inconsistent with respect to PL versus smooth, etc |
|
9 We expect that everything is true in the PL category, but at the moment our proof |
|
10 avails itself to smooth techniques. |
|
11 Furthermore, it traditional in the literature to speak of $C_*(\Diff(X))$ |
|
12 rather than $C_*(\Homeo(X))$.} |
|
13 |
5 |
14 |
6 Let $CH_*(X, Y)$ denote $C_*(\Homeo(X \to Y))$, the singular chain complex of |
15 Let $CH_*(X, Y)$ denote $C_*(\Homeo(X \to Y))$, the singular chain complex of |
7 the space of homeomorphisms |
16 the space of homeomorphisms |
8 between the $n$-manifolds $X$ and $Y$ (extending a fixed homeomorphism $\bd X \to \bd Y$). |
17 between the $n$-manifolds $X$ and $Y$ (extending a fixed homeomorphism $\bd X \to \bd Y$). |
9 For convenience, we will permit the singular cells generating $CH_*(X, Y)$ to be more general |
18 For convenience, we will permit the singular cells generating $CH_*(X, Y)$ to be more general |
77 |
86 |
78 Before diving into the details, we outline our strategy for the proof of Proposition \ref{CHprop}. |
87 Before diving into the details, we outline our strategy for the proof of Proposition \ref{CHprop}. |
79 |
88 |
80 %Suppose for the moment that evaluation maps with the advertised properties exist. |
89 %Suppose for the moment that evaluation maps with the advertised properties exist. |
81 Let $p$ be a singular cell in $CH_k(X)$ and $b$ be a blob diagram in $\bc_*(X)$. |
90 Let $p$ be a singular cell in $CH_k(X)$ and $b$ be a blob diagram in $\bc_*(X)$. |
82 Suppose that there exists $V \sub X$ such that |
91 We say that $p\ot b$ is {\it localizable} if there exists $V \sub X$ such that |
83 \begin{enumerate} |
92 \begin{itemize} |
84 \item $V$ is homeomorphic to a disjoint union of balls, and |
93 \item $V$ is homeomorphic to a disjoint union of balls, and |
85 \item $\supp(p) \cup \supp(b) \sub V$. |
94 \item $\supp(p) \cup \supp(b) \sub V$. |
86 \end{enumerate} |
95 \end{itemize} |
87 (Recall that $\supp(b)$ is defined to be the union of the blobs of the diagram $b$.) |
96 (Recall that $\supp(b)$ is defined to be the union of the blobs of the diagram $b$.) |
88 Let $W = X \setmin V$, $W' = p(W)$ and $V' = X\setmin W'$. |
97 |
|
98 Assuming that $p\ot b$ is localizable as above, |
|
99 let $W = X \setmin V$, $W' = p(W)$ and $V' = X\setmin W'$. |
89 We then have a factorization |
100 We then have a factorization |
90 \[ |
101 \[ |
91 p = \gl(q, r), |
102 p = \gl(q, r), |
92 \] |
103 \] |
93 where $q \in CH_k(V, V')$ and $r \in CH_0(W, W')$. |
104 where $q \in CH_k(V, V')$ and $r \in CH_0(W, W')$. |
111 \[ |
122 \[ |
112 \bd(e_{VV'}(q\otimes b_V)) = e_{VV'}(\bd(q\otimes b_V)) . |
123 \bd(e_{VV'}(q\otimes b_V)) = e_{VV'}(\bd(q\otimes b_V)) . |
113 \] |
124 \] |
114 |
125 |
115 Thus the conditions of the proposition determine (up to homotopy) the evaluation |
126 Thus the conditions of the proposition determine (up to homotopy) the evaluation |
116 map for generators $p\otimes b$ such that $\supp(p) \cup \supp(b)$ is contained in a disjoint |
127 map for localizable generators $p\otimes b$. |
117 union of balls. |
|
118 On the other hand, Lemma \ref{extension_lemma} allows us to homotope |
128 On the other hand, Lemma \ref{extension_lemma} allows us to homotope |
119 arbitrary generators to sums of generators with this property. |
129 arbitrary generators to sums of localizable generators. |
120 \nn{should give a name to this property; also forward reference} |
|
121 This (roughly) establishes the uniqueness part of the proposition. |
130 This (roughly) establishes the uniqueness part of the proposition. |
122 To show existence, we must show that the various choices involved in constructing |
131 To show existence, we must show that the various choices involved in constructing |
123 evaluation maps in this way affect the final answer only by a homotopy. |
132 evaluation maps in this way affect the final answer only by a homotopy. |
124 |
133 |
125 \nn{maybe put a little more into the outline before diving into the details.} |
134 Now for a little more detail. |
126 |
135 (But we're still just motivating the full, gory details, which will follow.) |
127 \noop{Note: At the moment this section is very inconsistent with respect to PL versus smooth, etc |
136 Choose a metric on $X$, and let $\cU_\gamma$ be the open cover of by balls of radius $\gamma$. |
128 We expect that everything is true in the PL category, but at the moment our proof |
137 By Lemma \ref{extension_lemma} we can restrict our attention to $k$-parameter families |
129 avails itself to smooth techniques. |
138 $p$ of homeomorphisms such that $\supp(p)$ is contained in the union of $k$ $\gamma$-balls. |
130 Furthermore, it traditional in the literature to speak of $C_*(\Diff(X))$ |
139 For fixed blob diagram $b$ and fixed $k$, it's not hard to show that for $\gamma$ small enough |
131 rather than $C_*(\Homeo(X))$.} |
140 $p\ot b$ must be localizable. |
|
141 On the other hand, for fixed $k$ and $\gamma$ there exist $p$ and $b$ such that $p\ot b$ is not localizable, |
|
142 and for fixed $\gamma$ and $b$ there exist non-localizable $p\ot b$ for sufficiently large $k$. |
|
143 Thus we will need to take an appropriate limit as $\gamma$ approaches zero. |
|
144 |
|
145 The construction of $e_X$, as outlined above, depends on various choices, one of which |
|
146 is the choice, for each localizable generator $p\ot b$, |
|
147 of disjoint balls $V$ containing $\supp(p)\cup\supp(b)$. |
|
148 Let $V'$ be another disjoint union of balls containing $\supp(p)\cup\supp(b)$, |
|
149 and assume that there exists yet another disjoint union of balls $W$ with $W$ containing |
|
150 $V\cup V'$. |
|
151 Then we can use $W$ to construct a homotopy between the two versions of $e_X$ |
|
152 associated to $V$ and $V'$. |
|
153 If we impose no constraints on $V$ and $V'$ then such a $W$ need not exist. |
|
154 Thus we will insist below that $V$ (and $V'$) be contained in small metric neighborhoods |
|
155 of $\supp(p)\cup\supp(b)$. |
|
156 Because we want not mere homotopy uniqueness but iterated homotopy uniqueness, |
|
157 we will similarly require that $W$ be contained in a slightly larger metric neighborhood of |
|
158 $\supp(p)\cup\supp(b)$, and so on. |
|
159 |
132 |
160 |
133 \medskip |
161 \medskip |
134 |
162 |
135 Now for the details. |
163 Now for the details. |
136 |
164 |