11 We also will use the abbreviated notation $CH_*(X) \deq CH_*(X, X)$. |
11 We also will use the abbreviated notation $CH_*(X) \deq CH_*(X, X)$. |
12 (For convenience, we will permit the singular cells generating $CH_*(X, Y)$ to be more general |
12 (For convenience, we will permit the singular cells generating $CH_*(X, Y)$ to be more general |
13 than simplices --- they can be based on any linear polyhedron. |
13 than simplices --- they can be based on any linear polyhedron. |
14 \nn{be more restrictive here? does more need to be said?}) |
14 \nn{be more restrictive here? does more need to be said?}) |
15 |
15 |
16 \begin{prop} \label{CHprop} |
16 \begin{thm} \label{thm:CH} |
17 For $n$-manifolds $X$ and $Y$ there is a chain map |
17 For $n$-manifolds $X$ and $Y$ there is a chain map |
18 \eq{ |
18 \eq{ |
19 e_{XY} : CH_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) |
19 e_{XY} : CH_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) |
20 } |
20 } |
21 such that |
21 such that |
22 \begin{enumerate} |
22 \begin{enumerate} |
23 \item on $CH_0(X, Y) \otimes \bc_*(X)$ it agrees with the obvious action of |
23 \item on $CH_0(X, Y) \otimes \bc_*(X)$ it agrees with the obvious action of |
24 $\Homeo(X, Y)$ on $\bc_*(X)$ (Property (\ref{property:functoriality})), and |
24 $\Homeo(X, Y)$ on $\bc_*(X)$ described in Property (\ref{property:functoriality}), and |
25 \item for any compatible splittings $X\to X\sgl$ and $Y\to Y\sgl$, |
25 \item for any compatible splittings $X\to X\sgl$ and $Y\to Y\sgl$, |
26 the following diagram commutes up to homotopy |
26 the following diagram commutes up to homotopy |
27 \begin{equation*} |
27 \begin{equation*} |
28 \xymatrix@C+2cm{ |
28 \xymatrix@C+2cm{ |
29 CH_*(X, Y) \otimes \bc_*(X) |
29 CH_*(X, Y) \otimes \bc_*(X) |
33 } |
33 } |
34 \end{equation*} |
34 \end{equation*} |
35 \end{enumerate} |
35 \end{enumerate} |
36 Up to (iterated) homotopy, there is a unique family $\{e_{XY}\}$ of chain maps |
36 Up to (iterated) homotopy, there is a unique family $\{e_{XY}\}$ of chain maps |
37 satisfying the above two conditions. |
37 satisfying the above two conditions. |
38 \end{prop} |
38 \end{thm} |
39 |
39 |
40 Before giving the proof, we state the essential technical tool of Lemma \ref{extension_lemma}, |
40 Before giving the proof, we state the essential technical tool of Lemma \ref{extension_lemma}, |
41 and then give an outline of the method of proof. |
41 and then give an outline of the method of proof. |
42 |
42 |
43 Without loss of generality, we will assume $X = Y$. |
43 Without loss of generality, we will assume $X = Y$. |
73 |
73 |
74 The proof will be given in \S\ref{sec:localising}. |
74 The proof will be given in \S\ref{sec:localising}. |
75 |
75 |
76 \medskip |
76 \medskip |
77 |
77 |
78 Before diving into the details, we outline our strategy for the proof of Proposition \ref{CHprop}. |
78 Before diving into the details, we outline our strategy for the proof of Theorem \ref{thm:CH}. |
79 Let $p$ be a singular cell in $CH_k(X)$ and $b$ be a blob diagram in $\bc_*(X)$. |
79 Let $p$ be a singular cell in $CH_k(X)$ and $b$ be a blob diagram in $\bc_*(X)$. |
80 We say that $p\ot b$ is {\it localizable} if there exists $V \sub X$ such that |
80 We say that $p\ot b$ is {\it localizable} if there exists $V \sub X$ such that |
81 \begin{itemize} |
81 \begin{itemize} |
82 \item $V$ is homeomorphic to a disjoint union of balls, and |
82 \item $V$ is homeomorphic to a disjoint union of balls, and |
83 \item $\supp(p) \cup \supp(b) \sub V$. |
83 \item $\supp(p) \cup \supp(b) \sub V$. |
145 Because we want not mere homotopy uniqueness but iterated homotopy uniqueness, |
145 Because we want not mere homotopy uniqueness but iterated homotopy uniqueness, |
146 we will similarly require that $W$ be contained in a slightly larger metric neighborhood of |
146 we will similarly require that $W$ be contained in a slightly larger metric neighborhood of |
147 $\supp(p)\cup\supp(b)$, and so on. |
147 $\supp(p)\cup\supp(b)$, and so on. |
148 |
148 |
149 |
149 |
150 \medskip |
150 \begin{proof}[Proof of Theorem \ref{thm:CH}.] |
151 |
|
152 \begin{proof}[Proof of Proposition \ref{CHprop}.] |
|
153 We'll use the notation $|b| = \supp(b)$ and $|p| = \supp(p)$. |
151 We'll use the notation $|b| = \supp(b)$ and $|p| = \supp(p)$. |
154 |
152 |
155 Choose a metric on $X$. |
153 Choose a metric on $X$. |
156 Choose a monotone decreasing sequence of positive real numbers $\ep_i$ converging to zero |
154 Choose a monotone decreasing sequence of positive real numbers $\ep_i$ converging to zero |
157 (e.g.\ $\ep_i = 2^{-i}$). |
155 (e.g.\ $\ep_i = 2^{-i}$). |
593 The gluing map $X\sgl\to X$ induces a map |
591 The gluing map $X\sgl\to X$ induces a map |
594 \[ |
592 \[ |
595 \gl: R_*\ot CH_*(X, X) \otimes \bc_*(X) \to R_*\ot CH_*(X\sgl, X \sgl) \otimes \bc_*(X \sgl) , |
593 \gl: R_*\ot CH_*(X, X) \otimes \bc_*(X) \to R_*\ot CH_*(X\sgl, X \sgl) \otimes \bc_*(X \sgl) , |
596 \] |
594 \] |
597 and it is easy to see that $\gl(G^m_*)\sub \ol{G}^m_*$. |
595 and it is easy to see that $\gl(G^m_*)\sub \ol{G}^m_*$. |
598 From this it follows that the diagram in the statement of Proposition \ref{CHprop} commutes. |
596 From this it follows that the diagram in the statement of Theorem \ref{thm:CH} commutes. |
599 |
597 |
600 \todo{this paragraph isn't very convincing, or at least I don't see what's going on} |
598 \todo{this paragraph isn't very convincing, or at least I don't see what's going on} |
601 Finally we show that the action maps defined above are independent of |
599 Finally we show that the action maps defined above are independent of |
602 the choice of metric (up to iterated homotopy). |
600 the choice of metric (up to iterated homotopy). |
603 The arguments are very similar to ones given above, so we only sketch them. |
601 The arguments are very similar to ones given above, so we only sketch them. |
612 Using acyclic models, we can construct a homotopy from $e$ to $e'$ on $F_*$. |
610 Using acyclic models, we can construct a homotopy from $e$ to $e'$ on $F_*$. |
613 We now observe that $CH_*(X, X) \ot \bc_*(X)$ retracts to $F_*$. |
611 We now observe that $CH_*(X, X) \ot \bc_*(X)$ retracts to $F_*$. |
614 Similar arguments show that this homotopy from $e$ to $e'$ is well-defined |
612 Similar arguments show that this homotopy from $e$ to $e'$ is well-defined |
615 up to second order homotopy, and so on. |
613 up to second order homotopy, and so on. |
616 |
614 |
617 This completes the proof of Proposition \ref{CHprop}. |
615 This completes the proof of Theorem \ref{thm:CH}. |
618 \end{proof} |
616 \end{proof} |
619 |
617 |
620 |
618 |
621 \begin{rem*} |
619 \begin{rem*} |
622 \label{rem:for-small-blobs} |
620 \label{rem:for-small-blobs} |
628 of $p(t,|b|)$ with some small balls. |
626 of $p(t,|b|)$ with some small balls. |
629 (Here ``small" is in terms of the metric on $X$ that we chose to construct $e$.) |
627 (Here ``small" is in terms of the metric on $X$ that we chose to construct $e$.) |
630 \end{rem*} |
628 \end{rem*} |
631 |
629 |
632 |
630 |
633 \begin{prop} |
631 \begin{thm} |
|
632 \label{thm:CH-associativity} |
634 The $CH_*(X, Y)$ actions defined above are associative. |
633 The $CH_*(X, Y)$ actions defined above are associative. |
635 That is, the following diagram commutes up to homotopy: |
634 That is, the following diagram commutes up to homotopy: |
636 \[ \xymatrix{ |
635 \[ \xymatrix{ |
637 & CH_*(Y, Z) \ot \bc_*(Y) \ar[dr]^{e_{YZ}} & \\ |
636 & CH_*(Y, Z) \ot \bc_*(Y) \ar[dr]^{e_{YZ}} & \\ |
638 CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X) \ar[ur]^{e_{XY}\ot\id} \ar[dr]_{\mu\ot\id} & & \bc_*(Z) \\ |
637 CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X) \ar[ur]^{e_{XY}\ot\id} \ar[dr]_{\mu\ot\id} & & \bc_*(Z) \\ |
639 & CH_*(X, Z) \ot \bc_*(X) \ar[ur]_{e_{XZ}} & |
638 & CH_*(X, Z) \ot \bc_*(X) \ar[ur]_{e_{XZ}} & |
640 } \] |
639 } \] |
641 Here $\mu:CH_*(X, Y) \ot CH_*(Y, Z)\to CH_*(X, Z)$ is the map induced by composition |
640 Here $\mu:CH_*(X, Y) \ot CH_*(Y, Z)\to CH_*(X, Z)$ is the map induced by composition |
642 of homeomorphisms. |
641 of homeomorphisms. |
643 \end{prop} |
642 \end{thm} |
644 |
643 |
645 \begin{proof} |
644 \begin{proof} |
646 The strategy of the proof is similar to that of Proposition \ref{CHprop}. |
645 The strategy of the proof is similar to that of Theorem \ref{thm:CH}. |
647 We will identify a subcomplex |
646 We will identify a subcomplex |
648 \[ |
647 \[ |
649 G_* \sub CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X) |
648 G_* \sub CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X) |
650 \] |
649 \] |
651 where it is easy to see that the two sides of the diagram are homotopic, then |
650 where it is easy to see that the two sides of the diagram are homotopic, then |
655 By definition, $p\ot q\ot b\in G_*$ if there is a disjoint union of balls in $X$ which |
654 By definition, $p\ot q\ot b\in G_*$ if there is a disjoint union of balls in $X$ which |
656 contains $|p| \cup p\inv(|q|) \cup |b|$. |
655 contains $|p| \cup p\inv(|q|) \cup |b|$. |
657 (If $p:P\times X\to Y$, then $p\inv(|q|)$ means the union over all $x\in P$ of |
656 (If $p:P\times X\to Y$, then $p\inv(|q|)$ means the union over all $x\in P$ of |
658 $p(x, \cdot)\inv(|q|)$.) |
657 $p(x, \cdot)\inv(|q|)$.) |
659 |
658 |
660 As in the proof of Proposition \ref{CHprop}, we can construct a homotopy |
659 As in the proof of Theorem \ref{thm:CH}, we can construct a homotopy |
661 between the upper and lower maps restricted to $G_*$. |
660 between the upper and lower maps restricted to $G_*$. |
662 This uses the facts that the maps agree on $CH_0(X, Y) \ot CH_0(Y, Z) \ot \bc_*(X)$, |
661 This uses the facts that the maps agree on $CH_0(X, Y) \ot CH_0(Y, Z) \ot \bc_*(X)$, |
663 that they are compatible with gluing, and the contractibility of $\bc_*(X)$. |
662 that they are compatible with gluing, and the contractibility of $\bc_*(X)$. |
664 |
663 |
665 We can now apply Lemma \ref{extension_lemma_c}, using a series of increasingly fine covers, |
664 We can now apply Lemma \ref{extension_lemma_c}, using a series of increasingly fine covers, |