78 \begin{lemma}[Small blobs] \label{small-blobs-b} \label{thm:small-blobs} |
78 \begin{lemma}[Small blobs] \label{small-blobs-b} \label{thm:small-blobs} |
79 The inclusion $i: \bc^{\cU}_*(X) \into \bc_*(X)$ is a homotopy equivalence. |
79 The inclusion $i: \bc^{\cU}_*(X) \into \bc_*(X)$ is a homotopy equivalence. |
80 \end{lemma} |
80 \end{lemma} |
81 |
81 |
82 \begin{proof} |
82 \begin{proof} |
83 It suffices \nn{why? we should spell this out somewhere} to show that for any finitely generated |
83 Since both complexes are free, it suffices to show that the inclusion induces |
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84 an isomorphism of homotopy groups. |
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85 To show that it suffices to show that for any finitely generated |
84 pair $(C_*, D_*)$, with $D_*$ a subcomplex of $C_*$ such that |
86 pair $(C_*, D_*)$, with $D_*$ a subcomplex of $C_*$ such that |
85 \[ |
87 \[ |
86 (C_*, D_*) \sub (\bc_*(X), \sbc_*(X)) |
88 (C_*, D_*) \sub (\bc_*(X), \sbc_*(X)) |
87 \] |
89 \] |
88 we can find a homotopy $h:C_*\to \bc_*(X)$ such that $h(D_*) \sub \sbc_*(X)$ |
90 we can find a homotopy $h:C_*\to \bc_*(X)$ such that $h(D_*) \sub \sbc_*(X)$ |
111 Roughly speaking, $s(b)$ consists of a series of 1-blob diagrams implementing a series |
113 Roughly speaking, $s(b)$ consists of a series of 1-blob diagrams implementing a series |
112 of small collar maps, plus a shrunken version of $b$. |
114 of small collar maps, plus a shrunken version of $b$. |
113 The composition of all the collar maps shrinks $B$ to a ball which is small with respect to $\cU$. |
115 The composition of all the collar maps shrinks $B$ to a ball which is small with respect to $\cU$. |
114 |
116 |
115 Let $\cV_1$ be an auxiliary open cover of $X$, subordinate to $\cU$ and |
117 Let $\cV_1$ be an auxiliary open cover of $X$, subordinate to $\cU$ and |
116 also satisfying conditions specified below. |
118 fine enough that a condition stated later in the proof is satisfied. |
117 Let $b = (B, u, r)$, with $u = \sum a_i$ the label of $B$, and $a_i\in \bc_0(B)$. |
119 Let $b = (B, u, r)$, with $u = \sum a_i$ the label of $B$, and $a_i\in \bc_0(B)$. |
118 Choose a sequence of collar maps $\bar{f}_j:B\cup\text{collar}\to B$ satisfying conditions which we cannot express |
120 Choose a sequence of collar maps $\bar{f}_j:B\cup\text{collar}\to B$ satisfying conditions |
119 until introducing more notation. \nn{needs some rewriting, I guess} |
121 specified at the end of this paragraph. |
120 Let $f_j:B\to B$ be the restriction of $\bar{f}_j$ to $B$; $f_j$ maps $B$ homeomorphically to |
122 Let $f_j:B\to B$ be the restriction of $\bar{f}_j$ to $B$; $f_j$ maps $B$ homeomorphically to |
121 a slightly smaller submanifold of $B$. |
123 a slightly smaller submanifold of $B$. |
122 Let $g_j = f_1\circ f_2\circ\cdots\circ f_j$. |
124 Let $g_j = f_1\circ f_2\circ\cdots\circ f_j$. |
123 Let $g$ be the last of the $g_j$'s. |
125 Let $g$ be the last of the $g_j$'s. |
124 Choose the sequence $\bar{f}_j$ so that |
126 Choose the sequence $\bar{f}_j$ so that |
125 $g(B)$ is contained is an open set of $\cV_1$ and |
127 $g(B)$ is contained is an open set of $\cV_1$ and |
126 $g_{j-1}(|f_j|)$ is also contained is an open set of $\cV_1$. |
128 $g_{j-1}(|f_j|)$ is also contained is an open set of $\cV_1$. |
127 |
129 |
128 There are 1-blob diagrams $c_{ij} \in \bc_1(B)$ such that $c_{ij}$ is compatible with $\cV_1$ |
130 There are 1-blob diagrams $c_{ij} \in \bc_1(B)$ such that $c_{ij}$ is compatible with $\cV_1$ |
129 (more specifically, $|c_{ij}| = g_{j-1}(|f_j|)$ \nn{doesn't strictly make any sense}) |
131 (more specifically, $|c_{ij}| = g_{j-1}(B)$) |
130 and $\bd c_{ij} = g_{j-1}(a_i) - g_{j}(a_i)$. |
132 and $\bd c_{ij} = g_{j-1}(a_i) - g_{j}(a_i)$. |
131 Define |
133 Define |
132 \[ |
134 \[ |
133 s(b) = \sum_{i,j} c_{ij} + g(b) |
135 s(b) = \sum_{i,j} c_{ij} + g(b) |
134 \] |
136 \] |
154 $s(b)$ consists of a series of 2-blob diagrams implementing a series |
156 $s(b)$ consists of a series of 2-blob diagrams implementing a series |
155 of small collar maps, plus a shrunken version of $b$. |
157 of small collar maps, plus a shrunken version of $b$. |
156 The composition of all the collar maps shrinks $B$ to a sufficiently small |
158 The composition of all the collar maps shrinks $B$ to a sufficiently small |
157 disjoint union of balls. |
159 disjoint union of balls. |
158 |
160 |
159 Let $\cV_2$ be an auxiliary open cover of $X$, subordinate to $\cU$ and |
161 Let $\cV_2$ be an auxiliary open cover of $X$, subordinate to $\cU$ and |
160 also satisfying conditions specified below. |
162 fine enough that a condition stated later in the proof is satisfied. |
161 \nn{This happens sufficiently far below (i.e. not in this paragraph) that we probably should give better warning.} |
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162 As before, choose a sequence of collar maps $f_j$ |
163 As before, choose a sequence of collar maps $f_j$ |
163 such that each has support |
164 such that each has support |
164 contained in an open set of $\cV_1$ and the composition of the corresponding collar homeomorphisms |
165 contained in an open set of $\cV_1$ and the composition of the corresponding collar homeomorphisms |
165 yields an embedding $g:B\to B$ such that $g(B)$ is contained in an open set of $\cV_1$. |
166 yields an embedding $g:B\to B$ such that $g(B)$ is contained in an open set of $\cV_1$. |
166 Let $g_j:B\to B$ be the embedding at the $j$-th stage. |
167 Let $g_j:B\to B$ be the embedding at the $j$-th stage. |
220 First we must specify a topology on the set of $k$-blob diagrams, $\BD_k$. |
221 First we must specify a topology on the set of $k$-blob diagrams, $\BD_k$. |
221 We give $\BD_k$ the finest topology such that |
222 We give $\BD_k$ the finest topology such that |
222 \begin{itemize} |
223 \begin{itemize} |
223 \item For any $b\in \BD_k$ the action map $\Homeo(X) \to \BD_k$, $f \mapsto f(b)$ is continuous. |
224 \item For any $b\in \BD_k$ the action map $\Homeo(X) \to \BD_k$, $f \mapsto f(b)$ is continuous. |
224 \item \nn{don't we need something for collaring maps?} |
225 \item \nn{don't we need something for collaring maps?} |
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226 \nn{KW: no, I don't think so. not unless we wanted some extension of $CH_*$ to act} |
225 \item The gluing maps $\BD_k(M)\to \BD_k(M\sgl)$ are continuous. |
227 \item The gluing maps $\BD_k(M)\to \BD_k(M\sgl)$ are continuous. |
226 \item For balls $B$, the map $U(B) \to \BD_1(B)$, $u\mapsto (B, u, \emptyset)$, is continuous, |
228 \item For balls $B$, the map $U(B) \to \BD_1(B)$, $u\mapsto (B, u, \emptyset)$, is continuous, |
227 where $U(B) \sub \bc_0(B)$ inherits its topology from $\bc_0(B)$ and the topology on |
229 where $U(B) \sub \bc_0(B)$ inherits its topology from $\bc_0(B)$ and the topology on |
228 $\bc_0(B)$ comes from the generating set $\BD_0(B)$. |
230 $\bc_0(B)$ comes from the generating set $\BD_0(B)$. |
229 \nn{don't we need to say more to specify a topology on an $\infty$-dimensional vector space} |
231 \nn{don't we need to say more to specify a topology on an $\infty$-dimensional vector space} |