189 It follows from Corollary \ref{disj-union-contract} that we can choose |
189 It follows from Corollary \ref{disj-union-contract} that we can choose |
190 $x_k \in \bc_2(X)$ with $\bd x_k = g_{j-1}(e_k) - g_j(e_k) - q(\bd e_k)$ |
190 $x_k \in \bc_2(X)$ with $\bd x_k = g_{j-1}(e_k) - g_j(e_k) - q(\bd e_k)$ |
191 and with $\supp(x_k) = U$. |
191 and with $\supp(x_k) = U$. |
192 We can now take $d_j \deq \sum x_k$. |
192 We can now take $d_j \deq \sum x_k$. |
193 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. |
193 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. |
194 \nn{should maybe have figure} |
194 \nn{should have figure} |
195 |
195 |
196 We now define |
196 We now define |
197 \[ |
197 \[ |
198 s(b) = \sum d_j + g(b), |
198 s(b) = \sum d_j + g(b), |
199 \] |
199 \] |
208 which contains finitely many open sets from $\cV_{l-1}$ |
208 which contains finitely many open sets from $\cV_{l-1}$ |
209 such that each ball is contained in some open set of $\cV_l$. |
209 such that each ball is contained in some open set of $\cV_l$. |
210 For sufficiently fine $\cV_{l-1}$ this will be possible. |
210 For sufficiently fine $\cV_{l-1}$ this will be possible. |
211 Since $C_*$ is finite, the process terminates after finitely many, say $r$, steps. |
211 Since $C_*$ is finite, the process terminates after finitely many, say $r$, steps. |
212 We take $\cV_r = \cU$. |
212 We take $\cV_r = \cU$. |
213 |
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214 \nn{should probably be more specific at the end} |
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215 \end{proof} |
213 \end{proof} |
216 |
214 |
217 |
215 |
218 \medskip |
216 \medskip |
219 |
217 |
220 Next we define the cone-product space version of the blob complex, $\btc_*(X)$. |
218 Next we define the cone-product space version of the blob complex, $\btc_*(X)$. |
221 First we must specify a topology on the set of $k$-blob diagrams, $\BD_k$. |
219 First we must specify a topology on the set of $k$-blob diagrams, $\BD_k$. |
222 We give $\BD_k$ the finest topology such that |
220 We give $\BD_k$ the finest topology such that |
223 \begin{itemize} |
221 \begin{itemize} |
224 \item For any $b\in \BD_k$ the action map $\Homeo(X) \to \BD_k$, $f \mapsto f(b)$ is continuous. |
222 \item For any $b\in \BD_k$ the action map $\Homeo(X) \to \BD_k$, $f \mapsto f(b)$ is continuous. |
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} |
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227 \item The gluing maps $\BD_k(M)\to \BD_k(M\sgl)$ are continuous. |
223 \item The gluing maps $\BD_k(M)\to \BD_k(M\sgl)$ are continuous. |
228 \item For balls $B$, the map $U(B) \to \BD_1(B)$, $u\mapsto (B, u, \emptyset)$, is continuous, |
224 \item For balls $B$, the map $U(B) \to \BD_1(B)$, $u\mapsto (B, u, \emptyset)$, is continuous, |
229 where $U(B) \sub \bc_0(B)$ inherits its topology from $\bc_0(B)$ and the topology on |
225 where $U(B) \sub \bc_0(B)$ inherits its topology from $\bc_0(B)$ and the topology on |
230 $\bc_0(B)$ comes from the generating set $\BD_0(B)$. |
226 $\bc_0(B)$ comes from the generating set $\BD_0(B)$. |
231 \nn{don't we need to say more to specify a topology on an $\infty$-dimensional vector space} |
227 \nn{don't we need to say more to specify a topology on an $\infty$-dimensional vector space} |
416 between the $n$-manifolds $X$ and $Y$ |
412 between the $n$-manifolds $X$ and $Y$ |
417 (any given singular chain extends a fixed homeomorphism $\bd X \to \bd Y$). |
413 (any given singular chain extends a fixed homeomorphism $\bd X \to \bd Y$). |
418 We also will use the abbreviated notation $CH_*(X) \deq CH_*(X, X)$. |
414 We also will use the abbreviated notation $CH_*(X) \deq CH_*(X, X)$. |
419 (For convenience, we will permit the singular cells generating $CH_*(X, Y)$ to be more general |
415 (For convenience, we will permit the singular cells generating $CH_*(X, Y)$ to be more general |
420 than simplices --- they can be based on any cone-product polyhedron (see Remark \ref{blobsset-remark}).) |
416 than simplices --- they can be based on any cone-product polyhedron (see Remark \ref{blobsset-remark}).) |
421 \nn{this note about our non-standard should probably go earlier in the paper, maybe intro} |
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422 |
417 |
423 \begin{thm} \label{thm:CH} |
418 \begin{thm} \label{thm:CH} |
424 For $n$-manifolds $X$ and $Y$ there is a chain map |
419 For $n$-manifolds $X$ and $Y$ there is a chain map |
425 \eq{ |
420 \eq{ |
426 e_{XY} : CH_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) , |
421 e_{XY} : CH_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) , |