6 In this section we extend the action of homeomorphisms on $\bc_*(X)$ |
6 In this section we extend the action of homeomorphisms on $\bc_*(X)$ |
7 to an action of {\it families} of homeomorphisms. |
7 to an action of {\it families} of homeomorphisms. |
8 That is, for each pair of homeomorphic manifolds $X$ and $Y$ |
8 That is, for each pair of homeomorphic manifolds $X$ and $Y$ |
9 we define a chain map |
9 we define a chain map |
10 \[ |
10 \[ |
11 e_{XY} : CH_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) , |
11 e_{XY} : \CH{X, Y} \otimes \bc_*(X) \to \bc_*(Y) , |
12 \] |
12 \] |
13 where $CH_*(X, Y) = C_*(\Homeo(X, Y))$, the singular chains on the space |
13 where $C_*(\Homeo(X, Y))$ is the singular chains on the space |
14 of homeomorphisms from $X$ to $Y$. |
14 of homeomorphisms from $X$ to $Y$. |
15 (If $X$ and $Y$ have non-empty boundary, these families of homeomorphisms |
15 (If $X$ and $Y$ have non-empty boundary, these families of homeomorphisms |
16 are required to restrict to a fixed homeomorphism on the boundaries.) |
16 are required to restrict to a fixed homeomorphism on the boundaries.) |
17 These actions (for various $X$ and $Y$) are compatible with gluing. |
17 These actions (for various $X$ and $Y$) are compatible with gluing. |
18 See \S \ref{ss:emap-def} for a more precise statement. |
18 See \S \ref{ss:emap-def} for a more precise statement. |
404 |
404 |
405 |
405 |
406 \subsection{Action of \texorpdfstring{$\CH{X}$}{C*(Homeo(M))}} |
406 \subsection{Action of \texorpdfstring{$\CH{X}$}{C*(Homeo(M))}} |
407 \label{ss:emap-def} |
407 \label{ss:emap-def} |
408 |
408 |
409 Let $CH_*(X, Y)$ denote $C_*(\Homeo(X \to Y))$, the singular chain complex of |
409 Let $C_*(\Homeo(X \to Y))$ denote the singular chain complex of |
410 the space of homeomorphisms |
410 the space of homeomorphisms |
411 between the $n$-manifolds $X$ and $Y$ |
411 between the $n$-manifolds $X$ and $Y$ |
412 (any given singular chain extends a fixed homeomorphism $\bd X \to \bd Y$). |
412 (any given singular chain extends a fixed homeomorphism $\bd X \to \bd Y$). |
413 We also will use the abbreviated notation $CH_*(X) \deq CH_*(X, X)$. |
413 We also will use the abbreviated notation $\CH{X} \deq \CH{X \to X}$. |
414 (For convenience, we will permit the singular cells generating $CH_*(X, Y)$ to be more general |
414 (For convenience, we will permit the singular cells generating $\CH{X \to Y}$ to be more general |
415 than simplices --- they can be based on any cone-product polyhedron (see Remark \ref{blobsset-remark}).) |
415 than simplices --- they can be based on any cone-product polyhedron (see Remark \ref{blobsset-remark}).) |
416 |
416 |
417 \begin{thm} \label{thm:CH} \label{thm:evaluation}% |
417 \begin{thm} \label{thm:CH} \label{thm:evaluation}% |
418 For $n$-manifolds $X$ and $Y$ there is a chain map |
418 For $n$-manifolds $X$ and $Y$ there is a chain map |
419 \eq{ |
419 \eq{ |
420 e_{XY} : CH_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) , |
420 e_{XY} : \CH{X \to Y} \otimes \bc_*(X) \to \bc_*(Y) , |
421 } |
421 } |
422 well-defined up to homotopy, |
422 well-defined up to homotopy, |
423 such that |
423 such that |
424 \begin{enumerate} |
424 \begin{enumerate} |
425 \item on $CH_0(X, Y) \otimes \bc_*(X)$ it agrees with the obvious action of |
425 \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 |
426 $\Homeo(X, Y)$ on $\bc_*(X)$ described in Property \ref{property:functoriality}, and |
427 \item for any compatible splittings $X\to X\sgl$ and $Y\to Y\sgl$, |
427 \item for any compatible splittings $X\to X\sgl$ and $Y\to Y\sgl$, |
428 the following diagram commutes up to homotopy |
428 the following diagram commutes up to homotopy |
429 \begin{equation*} |
429 \begin{equation*} |
430 \xymatrix@C+2cm{ |
430 \xymatrix@C+2cm{ |
431 CH_*(X, Y) \otimes \bc_*(X) |
431 \CH{X \to Y} \otimes \bc_*(X) |
432 \ar[r]_(.6){e_{XY}} \ar[d]^{\gl \otimes \gl} & |
432 \ar[r]_(.6){e_{XY}} \ar[d]^{\gl \otimes \gl} & |
433 \bc_*(Y)\ar[d]^{\gl} \\ |
433 \bc_*(Y)\ar[d]^{\gl} \\ |
434 CH_*(X\sgl, Y\sgl) \otimes \bc_*(X\sgl) \ar[r]_(.6){e_{X\sgl Y\sgl}} & \bc_*(Y\sgl) |
434 \CH{X\sgl, Y\sgl} \otimes \bc_*(X\sgl) \ar[r]_(.6){e_{X\sgl Y\sgl}} & \bc_*(Y\sgl) |
435 } |
435 } |
436 \end{equation*} |
436 \end{equation*} |
437 \end{enumerate} |
437 \end{enumerate} |
438 \end{thm} |
438 \end{thm} |
439 |
439 |
441 In light of Lemma \ref{lem:bc-btc}, it suffices to prove the theorem with |
441 In light of Lemma \ref{lem:bc-btc}, it suffices to prove the theorem with |
442 $\bc_*$ replaced by $\btc_*$. |
442 $\bc_*$ replaced by $\btc_*$. |
443 In fact, for $\btc_*$ we get a sharper result: we can omit |
443 In fact, for $\btc_*$ we get a sharper result: we can omit |
444 the ``up to homotopy" qualifiers. |
444 the ``up to homotopy" qualifiers. |
445 |
445 |
446 Let $f\in CH_k(X, Y)$, $f:P^k\to \Homeo(X \to Y)$ and $a\in \btc_{ij}(X)$, |
446 Let $f\in C_k(\Homeo(X \to Y))$, $f:P^k\to \Homeo(X \to Y)$ and $a\in \btc_{ij}(X)$, |
447 $a:Q^j \to \BD_i(X)$. |
447 $a:Q^j \to \BD_i(X)$. |
448 Define $e_{XY}(f\ot a)\in \btc_{i,j+k}(Y)$ by |
448 Define $e_{XY}(f\ot a)\in \btc_{i,j+k}(Y)$ by |
449 \begin{align*} |
449 \begin{align*} |
450 e_{XY}(f\ot a) : P\times Q &\to \BD_i(Y) \\ |
450 e_{XY}(f\ot a) : P\times Q &\to \BD_i(Y) \\ |
451 (p,q) &\mapsto f(p)(a(q)) . |
451 (p,q) &\mapsto f(p)(a(q)) . |
452 \end{align*} |
452 \end{align*} |
453 It is clear that this agrees with the previously defined $CH_0(X, Y)$ action on $\btc_*$, |
453 It is clear that this agrees with the previously defined $C_0(\Homeo(X \to Y))$ action on $\btc_*$, |
454 and it is also easy to see that the diagram in item 2 of the statement of the theorem |
454 and it is also easy to see that the diagram in item 2 of the statement of the theorem |
455 commutes on the nose. |
455 commutes on the nose. |
456 \end{proof} |
456 \end{proof} |
457 |
457 |
458 |
458 |
459 \begin{thm} |
459 \begin{thm} |
460 \label{thm:CH-associativity} |
460 \label{thm:CH-associativity} |
461 The $CH_*(X, Y)$ actions defined above are associative. |
461 The $\CH{X \to Y}$ actions defined above are associative. |
462 That is, the following diagram commutes up to homotopy: |
462 That is, the following diagram commutes up to homotopy: |
463 \[ \xymatrix{ |
463 \[ \xymatrix{ |
464 & CH_*(Y, Z) \ot \bc_*(Y) \ar[dr]^{e_{YZ}} & \\ |
464 & \CH{Y\to Z} \ot \bc_*(Y) \ar[dr]^{e_{YZ}} & \\ |
465 CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X) \ar[ur]^{e_{XY}\ot\id} \ar[dr]_{\mu\ot\id} & & \bc_*(Z) \\ |
465 \CH{X \to Y} \ot \CH{Y \to Z} \ot \bc_*(X) \ar[ur]^{e_{XY}\ot\id} \ar[dr]_{\mu\ot\id} & & \bc_*(Z) \\ |
466 & CH_*(X, Z) \ot \bc_*(X) \ar[ur]_{e_{XZ}} & |
466 & \CH{X \to Z} \ot \bc_*(X) \ar[ur]_{e_{XZ}} & |
467 } \] |
467 } \] |
468 Here $\mu:CH_*(X, Y) \ot CH_*(Y, Z)\to CH_*(X, Z)$ is the map induced by composition |
468 Here $\mu:\CH{X\to Y} \ot \CH{Y \to Z}\to \CH{X \to Z}$ is the map induced by composition |
469 of homeomorphisms. |
469 of homeomorphisms. |
470 \end{thm} |
470 \end{thm} |
471 \begin{proof} |
471 \begin{proof} |
472 The corresponding diagram for $\btc_*$ commutes on the nose. |
472 The corresponding diagram for $\btc_*$ commutes on the nose. |
473 \end{proof} |
473 \end{proof} |