47 Let $f: P \times X \to X$ be a family of homeomorphisms (e.g. a generator of $CH_*(X)$) |
47 Let $f: P \times X \to X$ be a family of homeomorphisms (e.g. a generator of $CH_*(X)$) |
48 and let $S \sub X$. |
48 and let $S \sub X$. |
49 We say that {\it $f$ is supported on $S$} if $f(p, x) = f(q, x)$ for all |
49 We say that {\it $f$ is supported on $S$} if $f(p, x) = f(q, x)$ for all |
50 $x \notin S$ 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' |
50 $x \notin S$ 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' |
51 homeomorphism $f_0 : X \to X$ so that |
51 homeomorphism $f_0 : X \to X$ so that |
52 \begin{align} |
52 \begin{align*} |
53 f(p,s) & = f_0(f'(p,s)) \;\;\;\; \mbox{for}\; (p, s) \in P\times S \\ |
53 f(p,s) & = f_0(f'(p,s)) \;\;\;\; \mbox{for}\; (p, s) \in P\times S \\ |
54 \intertext{and} |
54 \intertext{and} |
55 f(p,x) & = f_0(x) \;\;\;\; \mbox{for}\; (p, x) \in {P \times (X \setmin S)}. |
55 f(p,x) & = f_0(x) \;\;\;\; \mbox{for}\; (p, x) \in {P \times (X \setmin S)}. |
56 \end{align} |
56 \end{align*} |
57 Note that if $f$ is supported on $S$ then it is also supported on any $R \sup S$. |
57 Note that if $f$ is supported on $S$ then it is also supported on any $R \sup S$. |
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58 (So when we talk about ``the" support of a family, there is some ambiguity, |
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59 but this ambiguity will not matter to us.) |
58 |
60 |
59 Let $\cU = \{U_\alpha\}$ be an open cover of $X$. |
61 Let $\cU = \{U_\alpha\}$ be an open cover of $X$. |
60 A $k$-parameter family of homeomorphisms $f: P \times X \to X$ is |
62 A $k$-parameter family of homeomorphisms $f: P \times X \to X$ is |
61 {\it adapted to $\cU$} if there is a factorization |
63 {\it adapted to $\cU$} |
62 \eq{ |
64 \nn{or `weakly adapted'; need to decide on terminology} |
63 P = P_1 \times \cdots \times P_m |
65 if the support of $f$ is contained in the union |
64 } |
66 of at most $k$ of the $U_\alpha$'s. |
65 (for some $m \le k$) |
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66 and families of homeomorphisms |
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67 \eq{ |
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68 f_i : P_i \times X \to X |
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69 } |
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70 such that |
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71 \begin{itemize} |
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72 \item each $f_i$ is supported on some connected $V_i \sub X$; |
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73 \item the sets $V_i$ are mutually disjoint; |
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74 \item each $V_i$ is the union of at most $k_i$ of the $U_\alpha$'s, |
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75 where $k_i = \dim(P_i)$; and |
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76 \item $f(p, \cdot) = g \circ f_1(p_1, \cdot) \circ \cdots \circ f_m(p_m, \cdot)$ |
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77 for all $p = (p_1, \ldots, p_m)$, for some fixed $g \in \Homeo(X)$. |
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78 \end{itemize} |
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79 A chain $x \in CH_k(X)$ is (by definition) adapted to $\cU$ if it is the sum |
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80 of singular cells, each of which is adapted to $\cU$. |
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81 |
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82 (Actually, in this section we will only need families of homeomorphisms to be |
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83 {\it weakly adapted} to $\cU$, meaning that the support of $f$ is contained in the union |
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84 of at most $k$ of the $U_\alpha$'s.) |
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85 |
67 |
86 \begin{lemma} \label{extension_lemma} |
68 \begin{lemma} \label{extension_lemma} |
87 Let $x \in CH_k(X)$ be a singular chain such that $\bd x$ is adapted to $\cU$. |
69 Let $x \in CH_k(X)$ be a singular chain such that $\bd x$ is adapted to $\cU$. |
88 Then $x$ is homotopic (rel boundary) to some $x' \in CH_k(X)$ which is adapted to $\cU$. |
70 Then $x$ is homotopic (rel boundary) to some $x' \in CH_k(X)$ which is adapted to $\cU$. |
89 Furthermore, one can choose the homotopy so that its support is equal to the support of $x$. |
71 Furthermore, one can choose the homotopy so that its support is equal to the support of $x$. |
90 \end{lemma} |
72 \end{lemma} |
91 |
73 |
92 The proof will be given in Section \ref{sec:localising}. |
74 The proof will be given in Appendix \ref{sec:localising}. |
93 We will actually prove the following more general result. |
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94 Let $S$ and $T$ be an arbitrary topological spaces. |
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95 %\nn{might need to restrict $S$; the proof uses partition of unity on $S$; |
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96 %check this; or maybe just restrict the cover} |
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97 Let $CM_*(S, T)$ denote the singular chains on the space of continuous maps |
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98 from $S$ to $T$. |
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99 Let $\cU$ be an open cover of $S$ which affords a partition of unity. |
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100 \nn{for some $S$ and $\cU$ there is no partition of unity? like if $S$ is not paracompact? |
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101 in any case, in our applications $S$ will always be a manifold} |
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102 |
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103 \begin{lemma} \label{extension_lemma_b} |
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104 Let $x \in CM_k(S, T)$ be a singular chain such that $\bd x$ is adapted to $\cU$. |
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105 Then $x$ is homotopic (rel boundary) to some $x' \in CM_k(S, T)$ which is adapted to $\cU$. |
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106 Furthermore, one can choose the homotopy so that its support is equal to the support of $x$. |
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107 If $S$ and $T$ are manifolds, the statement remains true if we replace $CM_*(S, T)$ with |
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108 chains of smooth maps or immersions. |
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109 \end{lemma} |
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110 |
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111 |
75 |
112 \medskip |
76 \medskip |
113 |
77 |
114 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}. |
115 |
79 |
149 |
114 |
150 Thus the conditions of the proposition determine (up to homotopy) the evaluation |
115 Thus the conditions of the proposition determine (up to homotopy) the evaluation |
151 map for generators $p\otimes b$ such that $\supp(p) \cup \supp(b)$ is contained in a disjoint |
116 map for generators $p\otimes b$ such that $\supp(p) \cup \supp(b)$ is contained in a disjoint |
152 union of balls. |
117 union of balls. |
153 On the other hand, Lemma \ref{extension_lemma} allows us to homotope |
118 On the other hand, Lemma \ref{extension_lemma} allows us to homotope |
154 \nn{is this commonly used as a verb?} arbitrary generators to sums of generators with this property. |
119 arbitrary generators to sums of generators with this property. |
155 \nn{should give a name to this property; also forward reference} |
120 \nn{should give a name to this property; also forward reference} |
156 This (roughly) establishes the uniqueness part of the proposition. |
121 This (roughly) establishes the uniqueness part of the proposition. |
157 To show existence, we must show that the various choices involved in constructing |
122 To show existence, we must show that the various choices involved in constructing |
158 evaluation maps in this way affect the final answer only by a homotopy. |
123 evaluation maps in this way affect the final answer only by a homotopy. |
159 |
124 |
160 \nn{maybe put a little more into the outline before diving into the details.} |
125 \nn{maybe put a little more into the outline before diving into the details.} |
161 |
126 |
162 \nn{Note: At the moment this section is very inconsistent with respect to PL versus smooth, etc |
127 \noop{Note: At the moment this section is very inconsistent with respect to PL versus smooth, etc |
163 We expect that everything is true in the PL category, but at the moment our proof |
128 We expect that everything is true in the PL category, but at the moment our proof |
164 avails itself to smooth techniques. |
129 avails itself to smooth techniques. |
165 Furthermore, it traditional in the literature to speak of $C_*(\Diff(X))$ |
130 Furthermore, it traditional in the literature to speak of $C_*(\Diff(X))$ |
166 rather than $C_*(\Homeo(X))$.} |
131 rather than $C_*(\Homeo(X))$.} |
167 |
132 |