389 $s(b)\in \bc_2(X)$ with $\bd(s(b)) = \bd(b + h_1(\bd b))$ (by Corollary \ref{disj-union-contract}). |
389 $s(b)\in \bc_2(X)$ with $\bd(s(b)) = \bd(b + h_1(\bd b))$ (by Corollary \ref{disj-union-contract}). |
390 By Lemmas \ref{bt-contract} and \ref{btc-prod}, we can now find |
390 By Lemmas \ref{bt-contract} and \ref{btc-prod}, we can now find |
391 $h_2(b) \in \btc_3(X)$, also supported on $V$, such that $\bd(h_2(b)) = s(b) - b - h_1(\bd b)$ |
391 $h_2(b) \in \btc_3(X)$, also supported on $V$, such that $\bd(h_2(b)) = s(b) - b - h_1(\bd b)$ |
392 |
392 |
393 The general case, $h_k$, is similar. |
393 The general case, $h_k$, is similar. |
394 \end{proof} |
394 |
395 |
395 Note that it is possible to make the various choices above so that the homotopies we construct |
396 The proof of Lemma \ref{lem:bc-btc} constructs a homotopy inverse to the inclusion |
396 are fixed on $\bc_* \sub \btc_*$. |
397 $\bc_*(X)\sub \btc_*(X)$. |
397 It follows that we may assume that |
398 One might ask for more: a contractible set of possible homotopy inverses, or at least an |
398 the homotopy inverse to the inclusion constructed above is the identity on $\bc_*$. |
399 $m$-connected set for arbitrarily large $m$. |
399 Note that the complex of all homotopy inverses with this property is contractible, |
400 The latter can be achieved with finer control over the various |
400 so the homotopy inverse is well-defined up to a contractible set of choices. |
401 choices of disjoint unions of balls in the above proofs, but we will not pursue this here. |
401 \end{proof} |
|
402 |
|
403 %The proof of Lemma \ref{lem:bc-btc} constructs a homotopy inverse to the inclusion |
|
404 %$\bc_*(X)\sub \btc_*(X)$. |
|
405 %One might ask for more: a contractible set of possible homotopy inverses, or at least an |
|
406 %$m$-connected set for arbitrarily large $m$. |
|
407 %The latter can be achieved with finer control over the various |
|
408 %choices of disjoint unions of balls in the above proofs, but we will not pursue this here. |
402 |
409 |
403 |
410 |
404 |
411 |
405 |
412 |
406 \subsection{Action of \texorpdfstring{$\CH{X}$}{C*(Homeo(M))}} |
413 \subsection{Action of \texorpdfstring{$\CH{X}$}{C*(Homeo(M))}} |
417 \begin{thm} \label{thm:CH} \label{thm:evaluation}% |
424 \begin{thm} \label{thm:CH} \label{thm:evaluation}% |
418 For $n$-manifolds $X$ and $Y$ there is a chain map |
425 For $n$-manifolds $X$ and $Y$ there is a chain map |
419 \eq{ |
426 \eq{ |
420 e_{XY} : \CH{X \to Y} \otimes \bc_*(X) \to \bc_*(Y) , |
427 e_{XY} : \CH{X \to Y} \otimes \bc_*(X) \to \bc_*(Y) , |
421 } |
428 } |
422 well-defined up to homotopy, |
429 well-defined up to (coherent) homotopy, |
423 such that |
430 such that |
424 \begin{enumerate} |
431 \begin{enumerate} |
425 \item on $C_0(\Homeo(X \to Y)) \otimes \bc_*(X)$ it agrees with the obvious action of |
432 \item on $C_0(\Homeo(X \to Y)) \otimes \bc_*(X)$ it agrees with the obvious action of |
426 $\Homeo(X, Y)$ on $\bc_*(X)$ described in Property \ref{property:functoriality}, and |
433 $\Homeo(X, Y)$ on $\bc_*(X)$ described in Property \ref{property:functoriality}, and |
427 \item for any compatible splittings $X\to X\sgl$ and $Y\to Y\sgl$, |
434 \item for any compatible splittings $X\to X\sgl$ and $Y\to Y\sgl$, |
457 |
464 |
458 |
465 |
459 \begin{thm} |
466 \begin{thm} |
460 \label{thm:CH-associativity} |
467 \label{thm:CH-associativity} |
461 The $\CH{X \to Y}$ actions defined above are associative. |
468 The $\CH{X \to Y}$ actions defined above are associative. |
462 That is, the following diagram commutes up to homotopy: |
469 That is, the following diagram commutes up to coherent homotopy: |
463 \[ \xymatrix@C=5pt{ |
470 \[ \xymatrix@C=5pt{ |
464 & \CH{Y\to Z} \ot \bc_*(Y) \ar[drr]^{e_{YZ}} & &\\ |
471 & \CH{Y\to Z} \ot \bc_*(Y) \ar[drr]^{e_{YZ}} & &\\ |
465 \CH{X \to Y} \ot \CH{Y \to Z} \ot \bc_*(X) \ar[ur]^{e_{XY}\ot\id} \ar[dr]_{\mu\ot\id} & & & \bc_*(Z) \\ |
472 \CH{X \to Y} \ot \CH{Y \to Z} \ot \bc_*(X) \ar[ur]^{e_{XY}\ot\id} \ar[dr]_{\mu\ot\id} & & & \bc_*(Z) \\ |
466 & \CH{X \to Z} \ot \bc_*(X) \ar[urr]_{e_{XZ}} & & |
473 & \CH{X \to Z} \ot \bc_*(X) \ar[urr]_{e_{XZ}} & & |
467 } \] |
474 } \] |