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 pair $(C_*, D_*)$, with $D_*$ a subcomplex of $C_*$ such that |
83 It suffices \nn{why? we should spell this out somewhere} to show that for any finitely generated |
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84 pair $(C_*, D_*)$, with $D_*$ a subcomplex of $C_*$ such that |
84 \[ |
85 \[ |
85 (C_*, D_*) \sub (\bc_*(X), \sbc_*(X)) |
86 (C_*, D_*) \sub (\bc_*(X), \sbc_*(X)) |
86 \] |
87 \] |
87 we can find a homotopy $h:C_*\to \bc_*(X)$ such that $h(D_*) \sub \sbc_*(X)$ |
88 we can find a homotopy $h:C_*\to \bc_*(X)$ such that $h(D_*) \sub \sbc_*(X)$ |
88 and |
89 and |
154 of small collar maps, plus a shrunken version of $b$. |
155 of small collar maps, plus a shrunken version of $b$. |
155 The composition of all the collar maps shrinks $B$ to a sufficiently small |
156 The composition of all the collar maps shrinks $B$ to a sufficiently small |
156 disjoint union of balls. |
157 disjoint union of balls. |
157 |
158 |
158 Let $\cV_2$ be an auxiliary open cover of $X$, subordinate to $\cU$ and |
159 Let $\cV_2$ be an auxiliary open cover of $X$, subordinate to $\cU$ and |
159 also satisfying conditions specified below. \nn{This happens sufficiently far below (i.e. not in this paragraph) that we probably should give better warning.} |
160 also satisfying conditions specified below. |
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161 \nn{This happens sufficiently far below (i.e. not in this paragraph) that we probably should give better warning.} |
160 As before, choose a sequence of collar maps $f_j$ |
162 As before, choose a sequence of collar maps $f_j$ |
161 such that each has support |
163 such that each has support |
162 contained in an open set of $\cV_1$ and the composition of the corresponding collar homeomorphisms |
164 contained in an open set of $\cV_1$ and the composition of the corresponding collar homeomorphisms |
163 yields an embedding $g:B\to B$ such that $g(B)$ is contained in an open set of $\cV_1$. |
165 yields an embedding $g:B\to B$ such that $g(B)$ is contained in an open set of $\cV_1$. |
164 Let $g_j:B\to B$ be the embedding at the $j$-th stage. |
166 Let $g_j:B\to B$ be the embedding at the $j$-th stage. |
221 \item For any $b\in \BD_k$ the action map $\Homeo(X) \to \BD_k$, $f \mapsto f(b)$ is continuous. |
223 \item For any $b\in \BD_k$ the action map $\Homeo(X) \to \BD_k$, $f \mapsto f(b)$ is continuous. |
222 \item \nn{don't we need something for collaring maps?} |
224 \item \nn{don't we need something for collaring maps?} |
223 \item The gluing maps $\BD_k(M)\to \BD_k(M\sgl)$ are continuous. |
225 \item The gluing maps $\BD_k(M)\to \BD_k(M\sgl)$ are continuous. |
224 \item For balls $B$, the map $U(B) \to \BD_1(B)$, $u\mapsto (B, u, \emptyset)$, is continuous, |
226 \item For balls $B$, the map $U(B) \to \BD_1(B)$, $u\mapsto (B, u, \emptyset)$, is continuous, |
225 where $U(B) \sub \bc_0(B)$ inherits its topology from $\bc_0(B)$ and the topology on |
227 where $U(B) \sub \bc_0(B)$ inherits its topology from $\bc_0(B)$ and the topology on |
226 $\bc_0(B)$ comes from the generating set $\BD_0(B)$. \nn{don't we need to say more to specify a topology on an $\infty$-dimensional vector space} |
228 $\bc_0(B)$ comes from the generating set $\BD_0(B)$. |
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229 \nn{don't we need to say more to specify a topology on an $\infty$-dimensional vector space} |
227 \end{itemize} |
230 \end{itemize} |
228 |
231 |
229 We can summarize the above by saying that in the typical continuous family |
232 We can summarize the above by saying that in the typical continuous family |
230 $P\to \BD_k(X)$, $p\mapsto \left(B_i(p), u_i(p), r(p)\right)$, $B_i(p)$ and $r(p)$ are induced by a map |
233 $P\to \BD_k(X)$, $p\mapsto \left(B_i(p), u_i(p), r(p)\right)$, $B_i(p)$ and $r(p)$ are induced by a map |
231 $P\to \Homeo(X)$, with the twig blob labels $u_i(p)$ varying independently. |
234 $P\to \Homeo(X)$, with the twig blob labels $u_i(p)$ varying independently. |
268 \] |
271 \] |
269 adds an outermost blob, equal to all of $B^n$, to the $j$-parameter family of blob diagrams. |
272 adds an outermost blob, equal to all of $B^n$, to the $j$-parameter family of blob diagrams. |
270 Note that for fixed $i$, $e$ is a chain map, i.e. $\bd_t e = e \bd_t$. |
273 Note that for fixed $i$, $e$ is a chain map, i.e. $\bd_t e = e \bd_t$. |
271 |
274 |
272 A generator $y\in \btc_{0j}$ is a map $y:P\to \BD_0$, where $P$ is some $j$-dimensional polyhedron. |
275 A generator $y\in \btc_{0j}$ is a map $y:P\to \BD_0$, where $P$ is some $j$-dimensional polyhedron. |
273 We define $r(y)\in \btc_{0j}$ to be the constant function $r\circ y : P\to \BD_0$. \nn{I found it pretty confusing to reuse the letter $r$ here.} |
276 We define $r(y)\in \btc_{0j}$ to be the constant function $r\circ y : P\to \BD_0$. |
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277 \nn{I found it pretty confusing to reuse the letter $r$ here.} |
274 Let $c(r(y))\in \btc_{0,j+1}$ be the constant map from the cone of $P$ to $\BD_0$ taking |
278 Let $c(r(y))\in \btc_{0,j+1}$ be the constant map from the cone of $P$ to $\BD_0$ taking |
275 the same value (namely $r(y(p))$, for any $p\in P$). |
279 the same value (namely $r(y(p))$, for any $p\in P$). |
276 Let $e(y - r(y)) \in \btc_{1j}$ denote the $j$-parameter family of 1-blob diagrams |
280 Let $e(y - r(y)) \in \btc_{1j}$ denote the $j$-parameter family of 1-blob diagrams |
277 whose value at $p\in P$ is the blob $B^n$ with label $y(p) - r(y(p))$. |
281 whose value at $p\in P$ is the blob $B^n$ with label $y(p) - r(y(p))$. |
278 Now define, for $y\in \btc_{0j}$, |
282 Now define, for $y\in \btc_{0j}$, |
300 e(\bd_t x - r(\bd_t x)) + c(r(\bd_t x)) \\ |
304 e(\bd_t x - r(\bd_t x)) + c(r(\bd_t x)) \\ |
301 &= x - r(x) - \bd_t(c(r(x))) + c(r(\bd_t x)) \\ |
305 &= x - r(x) - \bd_t(c(r(x))) + c(r(\bd_t x)) \\ |
302 &= x - r(x) + r(x) \\ |
306 &= x - r(x) + r(x) \\ |
303 &= x. |
307 &= x. |
304 \end{align*} |
308 \end{align*} |
305 Here we have used the fact that $\bd_b(c(r(x))) = 0$ since $c(r(x))$ is a $0$-blob diagram, as well as that $\bd_t(e(r(x))) = e(r(\bd_t x))$ \nn{explain why this is true?} and $c(r(\bd_t x)) - \bd_t(c(r(x))) = r(x)$ \nn{explain?}. |
309 Here we have used the fact that $\bd_b(c(r(x))) = 0$ since $c(r(x))$ is a $0$-blob diagram, |
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310 as well as that $\bd_t(e(r(x))) = e(r(\bd_t x))$ |
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311 \nn{explain why this is true?} |
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312 and $c(r(\bd_t x)) - \bd_t(c(r(x))) = r(x)$ \nn{explain?}. |
306 |
313 |
307 For $x\in \btc_{00}$ we have |
314 For $x\in \btc_{00}$ we have |
308 \begin{align*} |
315 \begin{align*} |
309 \bd h(x) + h(\bd x) &= \bd_b(e(x - r(x))) + \bd_t(c(r(x))) \\ |
316 \bd h(x) + h(\bd x) &= \bd_b(e(x - r(x))) + \bd_t(c(r(x))) \\ |
310 &= x - r(x) + r(x) - r(x)\\ |
317 &= x - r(x) + r(x) - r(x)\\ |
515 CH_*(X\sgl, Y\sgl) \otimes \bc_*(X\sgl) \ar[r]_(.6){e_{X\sgl Y\sgl}} & \bc_*(Y\sgl) |
522 CH_*(X\sgl, Y\sgl) \otimes \bc_*(X\sgl) \ar[r]_(.6){e_{X\sgl Y\sgl}} & \bc_*(Y\sgl) |
516 } |
523 } |
517 \end{equation*} |
524 \end{equation*} |
518 \end{enumerate} |
525 \end{enumerate} |
519 Moreover, for any $m \geq 0$, we can find a family of chain maps $\{e_{XY}\}$ |
526 Moreover, for any $m \geq 0$, we can find a family of chain maps $\{e_{XY}\}$ |
520 satisfying the above two conditions which is $m$-connected. In particular, this means that the choice of chain map above is unique up to homotopy. |
527 satisfying the above two conditions which is $m$-connected. In particular, |
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528 this means that the choice of chain map above is unique up to homotopy. |
521 \end{thm} |
529 \end{thm} |
522 \begin{rem} |
530 \begin{rem} |
523 Note that the statement doesn't quite give uniqueness up to iterated homotopy. We fully expect that this should actually be the case, but haven't been able to prove this. |
531 Note that the statement doesn't quite give uniqueness up to iterated homotopy. |
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532 We fully expect that this should actually be the case, but haven't been able to prove this. |
524 \end{rem} |
533 \end{rem} |
525 |
534 |
526 |
535 |
527 Before giving the proof, we state the essential technical tool of Lemma \ref{extension_lemma}, |
536 Before giving the proof, we state the essential technical tool of Lemma \ref{extension_lemma}, |
528 and then give an outline of the method of proof. |
537 and then give an outline of the method of proof. |
710 near $\bd V$, the expressions $p(V) \sub X$ and $p(X\setmin V) \sub X$ are |
719 near $\bd V$, the expressions $p(V) \sub X$ and $p(X\setmin V) \sub X$ are |
711 unambiguous.) |
720 unambiguous.) |
712 We have $\deg(p'') = 0$ and, inductively, $f'' = p''(b'')$. |
721 We have $\deg(p'') = 0$ and, inductively, $f'' = p''(b'')$. |
713 %We also have that $\deg(b'') = 0 = \deg(p'')$. |
722 %We also have that $\deg(b'') = 0 = \deg(p'')$. |
714 Choose $x' \in \bc_*(p(V))$ such that $\bd x' = f'$. |
723 Choose $x' \in \bc_*(p(V))$ such that $\bd x' = f'$. |
715 This is possible by Properties \ref{property:disjoint-union} and \ref{property:contractibility} and the fact that isotopic fields |
724 This is possible by Properties \ref{property:disjoint-union} and \ref{property:contractibility} |
716 differ by a local relation. |
725 and the fact that isotopic fields differ by a local relation. |
717 Finally, define |
726 Finally, define |
718 \[ |
727 \[ |
719 e(p\ot b) \deq x' \bullet p''(b'') . |
728 e(p\ot b) \deq x' \bullet p''(b'') . |
720 \] |
729 \] |
721 |
730 |
827 for all $l\ge m$ and all $q\ot a$ which appear in the boundary of $p\ot b$. |
836 for all $l\ge m$ and all $q\ot a$ which appear in the boundary of $p\ot b$. |
828 |
837 |
829 |
838 |
830 \begin{proof} |
839 \begin{proof} |
831 |
840 |
832 There exists $\lambda > 0$ such that for every subset $c$ of the blobs of $b$ the set $\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < \lambda$ . |
841 There exists $\lambda > 0$ such that for every subset $c$ of the blobs of $b$ the set |
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842 $\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < \lambda$ . |
833 (Here we are using the fact that the blobs are |
843 (Here we are using the fact that the blobs are |
834 piecewise smooth or piecewise-linear and that $\bd c$ is collared.) |
844 piecewise smooth or piecewise-linear and that $\bd c$ is collared.) |
835 We need to consider all such $c$ because all generators appearing in |
845 We need to consider all such $c$ because all generators appearing in |
836 iterated boundaries of $p\ot b$ must be in $G_*^{i,m}$.) |
846 iterated boundaries of $p\ot b$ must be in $G_*^{i,m}$.) |
837 |
847 |