259 \begin{proof} |
259 \begin{proof} |
260 We will construct a contracting homotopy $h: \btc_*(B^n)\to \btc_{*+1}(B^n)$. |
260 We will construct a contracting homotopy $h: \btc_*(B^n)\to \btc_{*+1}(B^n)$. |
261 |
261 |
262 We will assume a splitting $s:H_0(\btc_*(B^n))\to \btc_0(B^n)$ |
262 We will assume a splitting $s:H_0(\btc_*(B^n))\to \btc_0(B^n)$ |
263 of the quotient map $q:\btc_0(B^n)\to H_0(\btc_*(B^n))$. |
263 of the quotient map $q:\btc_0(B^n)\to H_0(\btc_*(B^n))$. |
264 Let $r = s\circ q$. |
264 Let $\rho = s\circ q$. |
265 |
265 |
266 For $x\in \btc_{ij}$ with $i\ge 1$ define |
266 For $x\in \btc_{ij}$ with $i\ge 1$ define |
267 \[ |
267 \[ |
268 h(x) = e(x) , |
268 h(x) = e(x) , |
269 \] |
269 \] |
273 \] |
273 \] |
274 adds an outermost blob, equal to all of $B^n$, to the $j$-parameter family of blob diagrams. |
274 adds an outermost blob, equal to all of $B^n$, to the $j$-parameter family of blob diagrams. |
275 Note that for fixed $i$, $e$ is a chain map, i.e. $\bd_t e = e \bd_t$. |
275 Note that for fixed $i$, $e$ is a chain map, i.e. $\bd_t e = e \bd_t$. |
276 |
276 |
277 A generator $y\in \btc_{0j}$ is a map $y:P\to \BD_0$, where $P$ is some $j$-dimensional polyhedron. |
277 A generator $y\in \btc_{0j}$ is a map $y:P\to \BD_0$, where $P$ is some $j$-dimensional polyhedron. |
278 We define $r(y)\in \btc_{0j}$ to be the constant function $r\circ y : P\to \BD_0$. |
278 We define $r(y)\in \btc_{0j}$ to be the constant function $\rho\circ y : P\to \BD_0$. |
279 \nn{I found it pretty confusing to reuse the letter $r$ here.} |
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280 Let $c(r(y))\in \btc_{0,j+1}$ be the constant map from the cone of $P$ to $\BD_0$ taking |
279 Let $c(r(y))\in \btc_{0,j+1}$ be the constant map from the cone of $P$ to $\BD_0$ taking |
281 the same value (namely $r(y(p))$, for any $p\in P$). |
280 the same value (namely $r(y(p))$, for any $p\in P$). |
282 Let $e(y - r(y)) \in \btc_{1j}$ denote the $j$-parameter family of 1-blob diagrams |
281 Let $e(y - r(y)) \in \btc_{1j}$ denote the $j$-parameter family of 1-blob diagrams |
283 whose value at $p\in P$ is the blob $B^n$ with label $y(p) - r(y(p))$. |
282 whose value at $p\in P$ is the blob $B^n$ with label $y(p) - r(y(p))$. |
284 Now define, for $y\in \btc_{0j}$, |
283 Now define, for $y\in \btc_{0j}$, |
416 the space of homeomorphisms |
415 the space of homeomorphisms |
417 between the $n$-manifolds $X$ and $Y$ |
416 between the $n$-manifolds $X$ and $Y$ |
418 (any given singular chain extends a fixed homeomorphism $\bd X \to \bd Y$). |
417 (any given singular chain extends a fixed homeomorphism $\bd X \to \bd Y$). |
419 We also will use the abbreviated notation $CH_*(X) \deq CH_*(X, X)$. |
418 We also will use the abbreviated notation $CH_*(X) \deq CH_*(X, X)$. |
420 (For convenience, we will permit the singular cells generating $CH_*(X, Y)$ to be more general |
419 (For convenience, we will permit the singular cells generating $CH_*(X, Y)$ to be more general |
421 than simplices --- they can be based on any linear polyhedron.) |
420 than simplices --- they can be based on any cone-product polyhedron (see Remark \ref{blobsset-remark}).) |
422 \nn{be more restrictive here? (probably yes) does more need to be said?} |
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423 \nn{this note about our non-standard should probably go earlier in the paper, maybe intro} |
421 \nn{this note about our non-standard should probably go earlier in the paper, maybe intro} |
424 |
422 |
425 \begin{thm} \label{thm:CH} |
423 \begin{thm} \label{thm:CH} |
426 For $n$-manifolds $X$ and $Y$ there is a chain map |
424 For $n$-manifolds $X$ and $Y$ there is a chain map |
427 \eq{ |
425 \eq{ |