11 e_{XY} : CH_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) , |
11 e_{XY} : CH_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) , |
12 \] |
12 \] |
13 where $CH_*(X, Y) = C_*(\Homeo(X, Y))$, the singular chains on the space |
13 where $CH_*(X, Y) = C_*(\Homeo(X, Y))$, the singular chains on the space |
14 of homeomorphisms from $X$ to $Y$. |
14 of homeomorphisms from $X$ to $Y$. |
15 (If $X$ and $Y$ have non-empty boundary, these families of homeomorphisms |
15 (If $X$ and $Y$ have non-empty boundary, these families of homeomorphisms |
16 are required to be fixed on the boundaries.) |
16 are required to restrict to a fixed homeomorphism on the boundaries.) |
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17 These actions (for various $X$ and $Y$) are compatible with gluing. |
17 See \S \ref{ss:emap-def} for a more precise statement. |
18 See \S \ref{ss:emap-def} for a more precise statement. |
18 |
19 |
19 The most convenient way to prove that maps $e_{XY}$ with the desired properties exist is to |
20 The most convenient way to prove that maps $e_{XY}$ with the desired properties exist is to |
20 introduce a homotopy equivalent alternate version of the blob complex $\btc_*(X)$ |
21 introduce a homotopy equivalent alternate version of the blob complex, $\btc_*(X)$, |
21 which is more amenable to this sort of action. |
22 which is more amenable to this sort of action. |
22 Recall from Remark \ref{blobsset-remark} that blob diagrams |
23 Recall from Remark \ref{blobsset-remark} that blob diagrams |
23 have the structure of a sort-of-simplicial set. |
24 have the structure of a sort-of-simplicial set. |
24 Blob diagrams can also be equipped with a natural topology, which converts this |
25 Blob diagrams can also be equipped with a natural topology, which converts this |
25 sort-of-simplicial set into a sort-of-simplicial space. |
26 sort-of-simplicial set into a sort-of-simplicial space. |
26 Taking singular chains of this space we get $\btc_*(X)$. |
27 Taking singular chains of this space we get $\btc_*(X)$. |
27 The details are in \S \ref{ss:alt-def}. |
28 The details are in \S \ref{ss:alt-def}. |
28 For technical reasons we also show that requiring the blobs to be |
29 We also prove a useful lemma (\ref{small-blobs-b}) which says that we can assume that |
29 embedded yields a homotopy equivalent complex. |
30 blobs are small with respect to any fixed open cover. |
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31 |
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32 |
30 |
33 |
31 %Since $\bc_*(X)$ and $\btc_*(X)$ are homotopy equivalent one could try to construct |
34 %Since $\bc_*(X)$ and $\btc_*(X)$ are homotopy equivalent one could try to construct |
32 %the $CH_*$ actions directly in terms of $\bc_*(X)$. |
35 %the $CH_*$ actions directly in terms of $\bc_*(X)$. |
33 %This was our original approach, but working out the details created a nearly unreadable mess. |
36 %This was our original approach, but working out the details created a nearly unreadable mess. |
34 %We have salvaged a sketch of that approach in \S \ref{ss:old-evmap-remnants}. |
37 %We have salvaged a sketch of that approach in \S \ref{ss:old-evmap-remnants}. |
46 |
49 |
47 \medskip |
50 \medskip |
48 |
51 |
49 If $b$ is a blob diagram in $\bc_*(X)$, define the {\it support} of $b$, denoted |
52 If $b$ is a blob diagram in $\bc_*(X)$, define the {\it support} of $b$, denoted |
50 $\supp(b)$ or $|b|$, to be the union of the blobs of $b$. |
53 $\supp(b)$ or $|b|$, to be the union of the blobs of $b$. |
51 For a general $k$-chain $a\in \bc_k(X)$, define the support of $a$ to be the union |
54 %For a general $k$-chain $a\in \bc_k(X)$, define the support of $a$ to be the union |
52 of the supports of the blob diagrams which appear in it. |
55 %of the supports of the blob diagrams which appear in it. |
53 |
56 More generally, we say that a chain $a\in \bc_k(X)$ is supported on $S$ if |
54 If $f: P\times X\to X$ is a family of homeomorphisms and $Y\sub X$, we say that $f$ is |
57 $a = a'\bullet r$, where $a'\in \bc_k(S)$ and $r\in \bc_0(X\setmin S)$. |
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58 |
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59 Similarly, if $f: P\times X\to X$ is a family of homeomorphisms and $Y\sub X$, we say that $f$ is |
55 {\it supported on $Y$} if $f(p, x) = f(p', x)$ for all $x\in X\setmin Y$ and all $p,p'\in P$. |
60 {\it supported on $Y$} if $f(p, x) = f(p', x)$ for all $x\in X\setmin Y$ and all $p,p'\in P$. |
|
61 %Equivalently, $f = f'\bullet r$, where $f'\in CH_k(Y)$ and $r\in CH_0(X\setmin Y)$. |
56 We will sometimes abuse language and talk about ``the" support of $f$, |
62 We will sometimes abuse language and talk about ``the" support of $f$, |
57 again denoted $\supp(f)$ or $|f|$, to mean some particular choice of $Y$ such that |
63 again denoted $\supp(f)$ or $|f|$, to mean some particular choice of $Y$ such that |
58 $f$ is supported on $Y$. |
64 $f$ is supported on $Y$. |
59 |
65 |
60 If $f: M \cup (Y\times I) \to M$ is a collaring homeomorphism |
66 If $f: M \cup (Y\times I) \to M$ is a collaring homeomorphism |
61 (cf. end of \S \ref{ss:syst-o-fields}), |
67 (cf. end of \S \ref{ss:syst-o-fields}), |
62 we say that $f$ is supported on $S\sub M$ if $f(x) = x$ for all $x\in M\setmin S$. |
68 we say that $f$ is supported on $S\sub M$ if $f(x) = x$ for all $x\in M\setmin S$. |
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69 |
|
70 \medskip |
63 |
71 |
64 Fix $\cU$, an open cover of $X$. |
72 Fix $\cU$, an open cover of $X$. |
65 Define the ``small blob complex" $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(X)$ |
73 Define the ``small blob complex" $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(X)$ |
66 of all blob diagrams in which every blob is contained in some open set of $\cU$, |
74 of all blob diagrams in which every blob is contained in some open set of $\cU$, |
67 and moreover each field labeling a region cut out by the blobs is splittable |
75 and moreover each field labeling a region cut out by the blobs is splittable |
99 It then follows from \ref{disj-union-contract} that we can choose |
107 It then follows from \ref{disj-union-contract} that we can choose |
100 $h_1(b) \in \bc_1(X)$ such that $\bd(h_1(b)) = s(b) - b$. |
108 $h_1(b) \in \bc_1(X)$ such that $\bd(h_1(b)) = s(b) - b$. |
101 |
109 |
102 Roughly speaking, $s(b)$ consists of a series of 1-blob diagrams implementing a series |
110 Roughly speaking, $s(b)$ consists of a series of 1-blob diagrams implementing a series |
103 of small collar maps, plus a shrunken version of $b$. |
111 of small collar maps, plus a shrunken version of $b$. |
104 The composition of all the collar maps shrinks $B$ to a sufficiently small ball. |
112 The composition of all the collar maps shrinks $B$ to a ball which is small with respect to $\cU$. |
105 |
113 |
106 Let $\cV_1$ be an auxiliary open cover of $X$, subordinate to $\cU$ and |
114 Let $\cV_1$ be an auxiliary open cover of $X$, subordinate to $\cU$ and |
107 also satisfying conditions specified below. |
115 also satisfying conditions specified below. |
108 Let $b = (B, u, r)$, $u = \sum a_i$ be the label of $B$, $a_i\in \bc_0(B)$. |
116 Let $b = (B, u, r)$, $u = \sum a_i$ be the label of $B$, $a_i\in \bc_0(B)$. |
109 Choose a sequence of collar maps $f_j:\bc_0(B)\to\bc_0(B)$ such that each has support |
117 Choose a sequence of collar maps $\bar{f}_j:B\cup\text{collar}\to B$ satisfying conditions which we cannot express |
110 contained in an open set of $\cV_1$ and the composition of the corresponding collar homeomorphisms |
118 until introducing more notation. |
111 yields an embedding $g:B\to B$ such that $g(B)$ is contained in an open set of $\cV_1$. |
119 Let $f_j:B\to B$ be the restriction of $\bar{f}_j$ to $B$; $f_j$ maps $B$ homeomorphically to |
112 \nn{need to say this better; maybe give fig} |
120 a slightly smaller submanifold of $B$. |
113 Let $g_j:B\to B$ be the embedding at the $j$-th stage. |
121 Let $g_j = f_1\circ f_2\circ\cdots\circ f_j$. |
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122 Let $g$ be the last of the $g_j$'s. |
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123 Choose the sequence $\bar{f}_j$ so that |
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124 $g(B)$ is contained is an open set of $\cV_1$ and |
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125 $g_{j-1}(|f_j|)$ is also contained is an open set of $\cV_1$. |
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126 |
114 There are 1-blob diagrams $c_{ij} \in \bc_1(B)$ such that $c_{ij}$ is compatible with $\cV_1$ |
127 There are 1-blob diagrams $c_{ij} \in \bc_1(B)$ such that $c_{ij}$ is compatible with $\cV_1$ |
115 and $\bd c_{ij} = g_j(a_i) = g_{j-1}(a_i)$. |
128 (more specifically, $|c_{ij}| = g_{j-1}(|f_j|)$) |
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129 and $\bd c_{ij} = g_{j-1}(a_i) - g_{j}(a_i)$. |
116 Define |
130 Define |
117 \[ |
131 \[ |
118 s(b) = \sum_{i,j} c_{ij} + g(b) |
132 s(b) = \sum_{i,j} c_{ij} + g(b) |
119 \] |
133 \] |
120 and choose $h_1(b) \in \bc_1(X)$ such that |
134 and choose $h_1(b) \in \bc_1(X)$ such that |
148 contained in an open set of $\cV_1$ and the composition of the corresponding collar homeomorphisms |
162 contained in an open set of $\cV_1$ and the composition of the corresponding collar homeomorphisms |
149 yields an embedding $g:B\to B$ such that $g(B)$ is contained in an open set of $\cV_1$. |
163 yields an embedding $g:B\to B$ such that $g(B)$ is contained in an open set of $\cV_1$. |
150 Let $g_j:B\to B$ be the embedding at the $j$-th stage. |
164 Let $g_j:B\to B$ be the embedding at the $j$-th stage. |
151 |
165 |
152 Fix $j$. |
166 Fix $j$. |
153 We will construct a 2-chain $d_j$ such that $\bd(d_j) = g_j(s(\bd b)) - g_{j-1}(s(\bd b))$. |
167 We will construct a 2-chain $d_j$ such that $\bd d_j = g_{j-1}(s(\bd b)) - g_{j}(s(\bd b))$. |
154 Let $g_{j-1}(s(\bd b)) = \sum e_k$, and let $\{p_m\}$ be the 0-blob diagrams |
168 Let $s(\bd b) = \sum e_k$, and let $\{p_m\}$ be the 0-blob diagrams |
155 appearing in the boundaries of the $e_k$. |
169 appearing in the boundaries of the $e_k$. |
156 As in the construction of $h_1$, we can choose 1-blob diagrams $q_m$ such that |
170 As in the construction of $h_1$, we can choose 1-blob diagrams $q_m$ such that |
157 $\bd q_m = f_j(p_m) - p_m$ and $\supp(q_m)$ is contained in an open set of $\cV_1$. |
171 $\bd q_m = g_{j-1}(p_m) - g_j(p_m)$ and $\supp(q_m)$ is contained in an open set of $\cV_1$. |
158 %%% \nn{better not to do this, to make things more parallel with general case (?)} |
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159 %Furthermore, we can arrange that all of the $q_m$ have the same support, and that this support |
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160 %is contained in a open set of $\cV_1$. |
|
161 %(This is possible since there are only finitely many $p_m$.) |
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162 If $x$ is a sum of $p_m$'s, we denote the corresponding sum of $q_m$'s by $q(x)$. |
172 If $x$ is a sum of $p_m$'s, we denote the corresponding sum of $q_m$'s by $q(x)$. |
163 |
173 |
164 Now consider, for each $k$, $e_k + q(\bd e_k)$. |
174 Now consider, for each $k$, $g_{j-1}(e_k) - q(\bd e_k)$. |
165 This is a 1-chain whose boundary is $f_j(\bd e_k)$. |
175 This is a 1-chain whose boundary is $g_j(\bd e_k)$. |
166 The support of $e_k$ is $g_{j-1}(V)$ for some $V\in \cV_1$, and |
176 The support of $e_k$ is $g_{j-1}(V)$ for some $V\in \cV_1$, and |
167 the support of $q(\bd e_k)$ is contained in a union $V'$ of finitely many open sets |
177 the support of $q(\bd e_k)$ is contained in a union $V'$ of finitely many open sets |
168 of $\cV_1$, all of which contain the support of $f_j$. |
178 of $\cV_1$, all of which contain the support of $f_j$. |
169 %the support of $q(\bd e_k)$ is contained in $V'$ for some $V'\in \cV_1$. |
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170 We now reveal the mysterious condition (mentioned above) which $\cV_1$ satisfies: |
179 We now reveal the mysterious condition (mentioned above) which $\cV_1$ satisfies: |
171 the union of $g_{j-1}(V)$ and $V'$, for all of the finitely many instances |
180 the union of $g_{j-1}(V)$ and $V'$, for all of the finitely many instances |
172 arising in the construction of $h_2$, lies inside a disjoint union of balls $U$ |
181 arising in the construction of $h_2$, lies inside a disjoint union of balls $U$ |
173 such that each individual ball lies in an open set of $\cV_2$. |
182 such that each individual ball lies in an open set of $\cV_2$. |
174 (In this case there are either one or two balls in the disjoint union.) |
183 (In this case there are either one or two balls in the disjoint union.) |
175 For any fixed open cover $\cV_2$ this condition can be satisfied by choosing $\cV_1$ |
184 For any fixed open cover $\cV_2$ this condition can be satisfied by choosing $\cV_1$ |
176 to be a sufficiently fine cover. |
185 to be a sufficiently fine cover. |
177 It follows from \ref{disj-union-contract} |
186 It follows from \ref{disj-union-contract} that we can choose |
178 that we can choose $x_k \in \bc_2(X)$ with $\bd x_k = f_j(e_k) - e_k - q(\bd e_k)$ |
187 $x_k \in \bc_2(X)$ with $\bd x_k = g_{j-1}(e_k) - g_j(e_k) - q(\bd e_k)$ |
179 and with $\supp(x_k) = U$. |
188 and with $\supp(x_k) = U$. |
180 We can now take $d_j \deq \sum x_k$. |
189 We can now take $d_j \deq \sum x_k$. |
181 It is clear that $\bd d_j = \sum (f_j(e_k) - e_k) = g_j(s(\bd b)) - g_{j-1}(s(\bd b))$, as desired. |
190 It is clear that $\bd d_j = \sum (g_{j-1}(e_k) - g_j(e_k)) = g_{j-1}(s(\bd b)) - g_{j}(s(\bd b))$, as desired. |
182 \nn{should maybe have figure} |
191 \nn{should maybe have figure} |
183 |
192 |
184 We now define |
193 We now define |
185 \[ |
194 \[ |
186 s(b) = \sum d_j + g(b), |
195 s(b) = \sum d_j + g(b), |