85 (C_*, D_*) \sub (\bc_*(X), \sbc_*(X)) |
85 (C_*, D_*) \sub (\bc_*(X), \sbc_*(X)) |
86 \] |
86 \] |
87 we can find a homotopy $h:C_*\to \bc_*(X)$ such that $h(D_*) \sub \sbc_*(X)$ |
87 we can find a homotopy $h:C_*\to \bc_*(X)$ such that $h(D_*) \sub \sbc_*(X)$ |
88 and |
88 and |
89 \[ |
89 \[ |
90 h\bd(x) + \bd h(x) - x \in \sbc_*(X) |
90 h\bd(x) + \bd h(x) + x \in \sbc_*(X) |
91 \] |
91 \] |
92 for all $x\in C_*$. |
92 for all $x\in C_*$. |
93 |
93 |
94 For simplicity we will assume that all fields are splittable into small pieces, so that |
94 For simplicity we will assume that all fields are splittable into small pieces, so that |
95 $\sbc_0(X) = \bc_0$. |
95 $\sbc_0(X) = \bc_0$. |
217 Next we define the sort-of-simplicial space version of the blob complex, $\btc_*(X)$. |
217 Next we define the sort-of-simplicial space version of the blob complex, $\btc_*(X)$. |
218 First we must specify a topology on the set of $k$-blob diagrams, $\BD_k$. |
218 First we must specify a topology on the set of $k$-blob diagrams, $\BD_k$. |
219 We give $\BD_k$ the finest topology such that |
219 We give $\BD_k$ the finest topology such that |
220 \begin{itemize} |
220 \begin{itemize} |
221 \item For any $b\in \BD_k$ the action map $\Homeo(X) \to \BD_k$, $f \mapsto f(b)$ is continuous. |
221 \item For any $b\in \BD_k$ the action map $\Homeo(X) \to \BD_k$, $f \mapsto f(b)$ is continuous. |
222 \item \nn{something about blob labels and vector space structure} |
222 \item The gluing maps $\BD_k(M)\to \BD_k(M\sgl)$ are continuous. |
223 \item \nn{maybe also something about gluing} |
223 \item For balls $B$, the map $U(B) \to \BD_1(B)$, $u\mapsto (B, u, \emptyset)$, is continuous, |
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224 where $U(B) \sub \bc_0(B)$ inherits its topology from $\bc_0(B)$ and the topology on |
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225 $\bc_0(B)$ comes from the generating set $\BD_0(B)$. |
224 \end{itemize} |
226 \end{itemize} |
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227 |
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228 We can summarize the above by saying that in the typical continuous family |
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229 $P\to \BD_k(M)$, $p\mapsto (B_i(p), u_i(p), r(p)$, $B_i(p)$ and $r(p)$ are induced by a map |
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230 $P\to \Homeo(M)$, with the twig blob labels $u_i(p)$ varying independently. |
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231 We note that while have no need to allow the blobs $B_i(p)$ to vary independently of the field $r(p)$, |
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232 if we did allow this it would not affect the truth of the claims we make below. |
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233 In particular, we would get a homotopy equivalent complex $\btc_*(M)$. |
225 |
234 |
226 Next we define $\btc_*(X)$ to be the total complex of the double complex (denoted $\btc_{**}$) |
235 Next we define $\btc_*(X)$ to be the total complex of the double complex (denoted $\btc_{**}$) |
227 whose $(i,j)$ entry is $C_j(\BD_i)$, the singular $j$-chains on the space of $i$-blob diagrams. |
236 whose $(i,j)$ entry is $C_j(\BD_i)$, the singular $j$-chains on the space of $i$-blob diagrams. |
228 The vertical boundary of the double complex, |
237 The vertical boundary of the double complex, |
229 denoted $\bd_t$, is the singular boundary, and the horizontal boundary, denoted $\bd_b$, is |
238 denoted $\bd_t$, is the singular boundary, and the horizontal boundary, denoted $\bd_b$, is |
259 adds an outermost blob, equal to all of $B^n$, to the $j$-parameter family of blob diagrams. |
268 adds an outermost blob, equal to all of $B^n$, to the $j$-parameter family of blob diagrams. |
260 |
269 |
261 A generator $y\in \btc_{0j}$ is a map $y:P\to \BD_0$, where $P$ is some $j$-dimensional polyhedron. |
270 A generator $y\in \btc_{0j}$ is a map $y:P\to \BD_0$, where $P$ is some $j$-dimensional polyhedron. |
262 We define $r(y)\in \btc_{0j}$ to be the constant function $r\circ y : P\to \BD_0$. |
271 We define $r(y)\in \btc_{0j}$ to be the constant function $r\circ y : P\to \BD_0$. |
263 Let $c(r(y))\in \btc_{0,j+1}$ be the constant map from the cone of $P$ to $\BD_0$ taking |
272 Let $c(r(y))\in \btc_{0,j+1}$ be the constant map from the cone of $P$ to $\BD_0$ taking |
264 the same value (i.e.\ $r(y(p))$ for any $p\in P$). |
273 the same value (namely $r(y(p))$, for any $p\in P$). |
265 Let $e(y - r(y)) \in \btc_{1j}$ denote the $j$-parameter family of 1-blob diagrams |
274 Let $e(y - r(y)) \in \btc_{1j}$ denote the $j$-parameter family of 1-blob diagrams |
266 whose value at $p\in P$ is the blob $B^n$ with label $y(p) - r(y(p))$. |
275 whose value at $p\in P$ is the blob $B^n$ with label $y(p) - r(y(p))$. |
267 Now define, for $y\in \btc_{0j}$, |
276 Now define, for $y\in \btc_{0j}$, |
268 \[ |
277 \[ |
269 h(y) = e(y - r(y)) + c(r(y)) . |
278 h(y) = e(y - r(y)) + c(r(y)) . |
313 \BD_k(X\du Y) \cong \coprod_{i+j=k} \BD_i(X)\times\BD_j(Y) . |
322 \BD_k(X\du Y) \cong \coprod_{i+j=k} \BD_i(X)\times\BD_j(Y) . |
314 \] |
323 \] |
315 \end{proof} |
324 \end{proof} |
316 |
325 |
317 For $S\sub X$, we say that $a\in \btc_k(X)$ is {\it supported on $S$} |
326 For $S\sub X$, we say that $a\in \btc_k(X)$ is {\it supported on $S$} |
318 if there exists $S' \subeq S$, $a'\in \btc_k(S')$ |
327 if there exists $a'\in \btc_k(S)$ |
319 and $r\in \btc_0(X\setmin S')$ such that $a = a'\bullet r$. |
328 and $r\in \btc_0(X\setmin S)$ such that $a = a'\bullet r$. |
320 |
329 |
321 \newcommand\sbtc{\btc^{\cU}} |
330 \newcommand\sbtc{\btc^{\cU}} |
322 Let $\cU$ be an open cover of $X$. |
331 Let $\cU$ be an open cover of $X$. |
323 Let $\sbtc_*(X)\sub\btc_*(X)$ be the subcomplex generated by |
332 Let $\sbtc_*(X)\sub\btc_*(X)$ be the subcomplex generated by |
324 $a\in \btc_*(X)$ such that there is a decomposition $X = \cup_i D_i$ |
333 $a\in \btc_*(X)$ such that there is a decomposition $X = \cup_i D_i$ |
399 the space of homeomorphisms |
408 the space of homeomorphisms |
400 between the $n$-manifolds $X$ and $Y$ |
409 between the $n$-manifolds $X$ and $Y$ |
401 (any given singular chain extends a fixed homeomorphism $\bd X \to \bd Y$). |
410 (any given singular chain extends a fixed homeomorphism $\bd X \to \bd Y$). |
402 We also will use the abbreviated notation $CH_*(X) \deq CH_*(X, X)$. |
411 We also will use the abbreviated notation $CH_*(X) \deq CH_*(X, X)$. |
403 (For convenience, we will permit the singular cells generating $CH_*(X, Y)$ to be more general |
412 (For convenience, we will permit the singular cells generating $CH_*(X, Y)$ to be more general |
404 than simplices --- they can be based on any linear polyhedron. |
413 than simplices --- they can be based on any linear polyhedron.) |
405 \nn{be more restrictive here? does more need to be said?}) |
414 \nn{be more restrictive here? (probably yes) does more need to be said?} |
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415 \nn{this note about our non-standard should probably go earlier in the paper, maybe intro} |
406 |
416 |
407 \begin{thm} \label{thm:CH} |
417 \begin{thm} \label{thm:CH} |
408 For $n$-manifolds $X$ and $Y$ there is a chain map |
418 For $n$-manifolds $X$ and $Y$ there is a chain map |
409 \eq{ |
419 \eq{ |
410 e_{XY} : CH_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) , |
420 e_{XY} : CH_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) , |