34 Taking singular chains of this space we get $\btc_*(X)$. |
26 Taking singular chains of this space we get $\btc_*(X)$. |
35 The details are in \S \ref{ss:alt-def}. |
27 The details are in \S \ref{ss:alt-def}. |
36 For technical reasons we also show that requiring the blobs to be |
28 For technical reasons we also show that requiring the blobs to be |
37 embedded yields a homotopy equivalent complex. |
29 embedded yields a homotopy equivalent complex. |
38 |
30 |
39 Since $\bc_*(X)$ and $\btc_*(X)$ are homotopy equivalent one could try to construct |
31 %Since $\bc_*(X)$ and $\btc_*(X)$ are homotopy equivalent one could try to construct |
40 the $CH_*$ actions directly in terms of $\bc_*(X)$. |
32 %the $CH_*$ actions directly in terms of $\bc_*(X)$. |
41 This was our original approach, but working out the details created a nearly unreadable mess. |
33 %This was our original approach, but working out the details created a nearly unreadable mess. |
42 We have salvaged a sketch of that approach in \S \ref{ss:old-evmap-remnants}. |
34 %We have salvaged a sketch of that approach in \S \ref{ss:old-evmap-remnants}. |
43 |
35 % |
44 \nn{should revisit above intro after this section is done} |
36 %\nn{should revisit above intro after this section is done} |
45 |
37 |
46 |
38 |
47 \subsection{Alternative definitions of the blob complex} |
39 \subsection{Alternative definitions of the blob complex} |
48 \label{ss:alt-def} |
40 \label{ss:alt-def} |
49 |
41 |
73 Define the ``small blob complex" $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(X)$ |
65 Define the ``small blob complex" $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(X)$ |
74 of all blob diagrams in which every blob is contained in some open set of $\cU$, |
66 of all blob diagrams in which every blob is contained in some open set of $\cU$, |
75 and moreover each field labeling a region cut out by the blobs is splittable |
67 and moreover each field labeling a region cut out by the blobs is splittable |
76 into fields on smaller regions, each of which is contained in some open set of $\cU$. |
68 into fields on smaller regions, each of which is contained in some open set of $\cU$. |
77 |
69 |
78 \begin{lemma}[Small blobs] \label{small-blobs-b} |
70 \begin{lemma}[Small blobs] \label{small-blobs-b} \label{thm:small-blobs} |
79 The inclusion $i: \bc^{\cU}_*(M) \into \bc_*(M)$ is a homotopy equivalence. |
71 The inclusion $i: \bc^{\cU}_*(M) \into \bc_*(M)$ is a homotopy equivalence. |
80 \end{lemma} |
72 \end{lemma} |
81 |
73 |
82 \begin{proof} |
74 \begin{proof} |
83 It suffices to show that for any finitely generated pair of subcomplexes |
75 It suffices to show that for any finitely generated pair of subcomplexes |
84 $(C_*, D_*) \sub (\bc_*(X), \sbc_*(X))$ |
76 \[ |
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77 (C_*, D_*) \sub (\bc_*(X), \sbc_*(X)) |
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78 \] |
85 we can find a homotopy $h:C_*\to \bc_*(X)$ such that $h(D_*) \sub \sbc_*(X)$ |
79 we can find a homotopy $h:C_*\to \bc_*(X)$ such that $h(D_*) \sub \sbc_*(X)$ |
86 and $x + h\bd(x) + \bd h(X) \in \sbc_*(X)$ for all $x\in C_*$. |
80 and |
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81 \[ |
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82 x + h\bd(x) + \bd h(X) \in \sbc_*(X) |
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83 \] |
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84 for all $x\in C_*$. |
87 |
85 |
88 For simplicity we will assume that all fields are splittable into small pieces, so that |
86 For simplicity we will assume that all fields are splittable into small pieces, so that |
89 $\sbc_0(X) = \bc_0$. |
87 $\sbc_0(X) = \bc_0$. |
90 (This is true for all of the examples presented in this paper.) |
88 (This is true for all of the examples presented in this paper.) |
91 Accordingly, we define $h_0 = 0$. |
89 Accordingly, we define $h_0 = 0$. |
355 We choose $\cU$ fine enough so that each generator of $C_*$ is supported on a disjoint union of balls. |
353 We choose $\cU$ fine enough so that each generator of $C_*$ is supported on a disjoint union of balls. |
356 (This is possible since the original $C_*$ was finite and therefore had bounded dimension.) |
354 (This is possible since the original $C_*$ was finite and therefore had bounded dimension.) |
357 |
355 |
358 Since $\bc_0(X) = \btc_0(X)$, we can take $h_0 = 0$. |
356 Since $\bc_0(X) = \btc_0(X)$, we can take $h_0 = 0$. |
359 |
357 |
360 |
358 Let $b \in C_1$ be a generator. |
361 |
359 Since $b$ is supported in a disjoint union of balls, |
362 |
360 we can find $s(b)\in \bc_1$ with $\bd (s(b)) = \bd b$ |
363 \nn{...} |
361 (by \ref{disj-union-contract}), and also $h_1(b) \in \btc_2$ |
364 \end{proof} |
362 such that $\bd (h_1(b)) = s(b) - b$ |
365 |
363 (by \ref{bt-contract} and \ref{btc-prod}). |
366 |
364 |
367 |
365 Now let $b$ be a generator of $C_2$. |
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366 If $\cU$ is fine enough, there is a disjoint union of balls $V$ |
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367 on which $b + h_1(\bd b)$ is supported. |
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368 Since $\bd(b + h_1(\bd b)) = s(\bd b) \in \bc_2$, we can find |
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369 $s(b)\in \bc_2$ with $\bd(s(b)) = \bd(b + h_1(\bd b))$ (by \ref{disj-union-contract}). |
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370 By \ref{bt-contract} and \ref{btc-prod}, we can now find |
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371 $h_2(b) \in \btc_3$, also supported on $V$, such that $\bd(h_2(b)) = s(b) - b - h_1(\bd b)$ |
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372 |
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373 The general case, $h_k$, is similar. |
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374 \end{proof} |
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375 |
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376 The proof of \ref{lem:bc-btc} constructs a homotopy inverse to the inclusion |
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377 $\bc_*(X)\sub \btc_*(X)$. |
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378 One might ask for more: a contractible set of possible homotopy inverses, or at least an |
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379 $m$-connected set for arbitrarily large $m$. |
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380 The latter can be achieved with finer control over the various |
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381 choices of disjoint unions of balls in the above proofs, but we will not pursue this here. |
368 |
382 |
369 |
383 |
370 |
384 |
371 |
385 |
372 \subsection{Action of \texorpdfstring{$\CH{X}$}{C*(Homeo(M))}} |
386 \subsection{Action of \texorpdfstring{$\CH{X}$}{C*(Homeo(M))}} |
373 \label{ss:emap-def} |
387 \label{ss:emap-def} |
374 |
388 |
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389 Let $CH_*(X, Y)$ denote $C_*(\Homeo(X \to Y))$, the singular chain complex of |
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390 the space of homeomorphisms |
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391 between the $n$-manifolds $X$ and $Y$ |
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392 (any given singular chain extends a fixed homeomorphism $\bd X \to \bd Y$). |
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393 We also will use the abbreviated notation $CH_*(X) \deq CH_*(X, X)$. |
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394 (For convenience, we will permit the singular cells generating $CH_*(X, Y)$ to be more general |
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395 than simplices --- they can be based on any linear polyhedron. |
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396 \nn{be more restrictive here? does more need to be said?}) |
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397 |
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398 \begin{thm} \label{thm:CH} |
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399 For $n$-manifolds $X$ and $Y$ there is a chain map |
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400 \eq{ |
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401 e_{XY} : CH_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) , |
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402 } |
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403 well-defined up to homotopy, |
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404 such that |
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405 \begin{enumerate} |
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406 \item on $CH_0(X, Y) \otimes \bc_*(X)$ it agrees with the obvious action of |
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407 $\Homeo(X, Y)$ on $\bc_*(X)$ described in Property (\ref{property:functoriality}), and |
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408 \item for any compatible splittings $X\to X\sgl$ and $Y\to Y\sgl$, |
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409 the following diagram commutes up to homotopy |
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410 \begin{equation*} |
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411 \xymatrix@C+2cm{ |
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412 CH_*(X, Y) \otimes \bc_*(X) |
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413 \ar[r]_(.6){e_{XY}} \ar[d]^{\gl \otimes \gl} & |
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414 \bc_*(Y)\ar[d]^{\gl} \\ |
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415 CH_*(X\sgl, Y\sgl) \otimes \bc_*(X\sgl) \ar[r]_(.6){e_{X\sgl Y\sgl}} & \bc_*(Y\sgl) |
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416 } |
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417 \end{equation*} |
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418 \end{enumerate} |
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419 \end{thm} |
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420 |
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421 \begin{proof} |
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422 In light of Lemma \ref{lem:bc-btc}, it suffices to prove the theorem with |
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423 $\bc_*$ replaced by $\btc_*$. |
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424 And in fact for $\btc_*$ we get a sharper result: we can omit |
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425 the ``up to homotopy" qualifiers. |
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426 |
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427 Let $f\in CH_k(X, Y)$, $f:P^k\to \Homeo(X \to Y)$ and $a\in \btc_{ij}(X)$, |
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428 $a:Q^j \to \BD_i(X)$. |
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429 Define $e_{XY}(f\ot a)\in \btc_{i,j+k}(Y)$ by |
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430 \begin{align*} |
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431 e_{XY}(f\ot a) : P\times Q &\to \BD_i(Y) \\ |
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432 (p,q) &\mapsto f(p)(a(q)) . |
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433 \end{align*} |
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434 It is clear that this agrees with the previously defined $CH_0(X, Y)$ action on $\btc_*$, |
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435 and it is also easy to see that the diagram in item 2 of the statement of the theorem |
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436 commutes on the nose. |
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437 \end{proof} |
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438 |
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439 |
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440 \begin{thm} |
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441 \label{thm:CH-associativity} |
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442 The $CH_*(X, Y)$ actions defined above are associative. |
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443 That is, the following diagram commutes up to homotopy: |
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444 \[ \xymatrix{ |
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445 & CH_*(Y, Z) \ot \bc_*(Y) \ar[dr]^{e_{YZ}} & \\ |
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446 CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X) \ar[ur]^{e_{XY}\ot\id} \ar[dr]_{\mu\ot\id} & & \bc_*(Z) \\ |
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447 & CH_*(X, Z) \ot \bc_*(X) \ar[ur]_{e_{XZ}} & |
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448 } \] |
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449 Here $\mu:CH_*(X, Y) \ot CH_*(Y, Z)\to CH_*(X, Z)$ is the map induced by composition |
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450 of homeomorphisms. |
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451 \end{thm} |
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452 \begin{proof} |
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453 The corresponding diagram for $\btc_*$ commutes on the nose. |
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454 \end{proof} |
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455 |
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456 |
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457 |
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458 |
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459 |
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460 |
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461 |
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462 \noop{ |
375 |
463 |
376 |
464 |
377 \subsection{[older version still hanging around]} |
465 \subsection{[older version still hanging around]} |
378 \label{ss:old-evmap-remnants} |
466 \label{ss:old-evmap-remnants} |
379 |
467 |