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 monotonically to zero |
154 Choose a monotone decreasing sequence of positive real numbers $\ep_i$ converging monotonically to zero |
157 (e.g.\ $\ep_i = 2^{-i}$). |
155 (e.g.\ $\ep_i = 2^{-i}$). |
592 The gluing map $X\sgl\to X$ induces a map |
590 The gluing map $X\sgl\to X$ induces a map |
593 \[ |
591 \[ |
594 \gl: R_*\ot CH_*(X, X) \otimes \bc_*(X) \to R_*\ot CH_*(X\sgl, X \sgl) \otimes \bc_*(X \sgl) , |
592 \gl: R_*\ot CH_*(X, X) \otimes \bc_*(X) \to R_*\ot CH_*(X\sgl, X \sgl) \otimes \bc_*(X \sgl) , |
595 \] |
593 \] |
596 and it is easy to see that $\gl(G^m_*)\sub \ol{G}^m_*$. |
594 and it is easy to see that $\gl(G^m_*)\sub \ol{G}^m_*$. |
597 From this it follows that the diagram in the statement of Proposition \ref{CHprop} commutes. |
595 From this it follows that the diagram in the statement of Theorem \ref{thm:CH} commutes. |
598 |
596 |
599 \todo{this paragraph isn't very convincing, or at least I don't see what's going on} |
597 \todo{this paragraph isn't very convincing, or at least I don't see what's going on} |
600 Finally we show that the action maps defined above are independent of |
598 Finally we show that the action maps defined above are independent of |
601 the choice of metric (up to iterated homotopy). |
599 the choice of metric (up to iterated homotopy). |
602 The arguments are very similar to ones given above, so we only sketch them. |
600 The arguments are very similar to ones given above, so we only sketch them. |
611 Using acyclic models, we can construct a homotopy from $e$ to $e'$ on $F_*$. |
609 Using acyclic models, we can construct a homotopy from $e$ to $e'$ on $F_*$. |
612 We now observe that $CH_*(X, X) \ot \bc_*(X)$ retracts to $F_*$. |
610 We now observe that $CH_*(X, X) \ot \bc_*(X)$ retracts to $F_*$. |
613 Similar arguments show that this homotopy from $e$ to $e'$ is well-defined |
611 Similar arguments show that this homotopy from $e$ to $e'$ is well-defined |
614 up to second order homotopy, and so on. |
612 up to second order homotopy, and so on. |
615 |
613 |
616 This completes the proof of Proposition \ref{CHprop}. |
614 This completes the proof of Theorem \ref{thm:CH}. |
617 \end{proof} |
615 \end{proof} |
618 |
616 |
619 |
617 |
620 \begin{rem*} |
618 \begin{rem*} |
621 \label{rem:for-small-blobs} |
619 \label{rem:for-small-blobs} |
627 of $p(t,|b|)$ with some small balls. |
625 of $p(t,|b|)$ with some small balls. |
628 (Here ``small" is in terms of the metric on $X$ that we chose to construct $e$.) |
626 (Here ``small" is in terms of the metric on $X$ that we chose to construct $e$.) |
629 \end{rem*} |
627 \end{rem*} |
630 |
628 |
631 |
629 |
632 \begin{prop} |
630 \begin{thm} |
|
631 \label{thm:CH-associativity} |
633 The $CH_*(X, Y)$ actions defined above are associative. |
632 The $CH_*(X, Y)$ actions defined above are associative. |
634 That is, the following diagram commutes up to homotopy: |
633 That is, the following diagram commutes up to homotopy: |
635 \[ \xymatrix{ |
634 \[ \xymatrix{ |
636 & CH_*(Y, Z) \ot \bc_*(Y) \ar[dr]^{e_{YZ}} & \\ |
635 & CH_*(Y, Z) \ot \bc_*(Y) \ar[dr]^{e_{YZ}} & \\ |
637 CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X) \ar[ur]^{e_{XY}\ot\id} \ar[dr]_{\mu\ot\id} & & \bc_*(Z) \\ |
636 CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X) \ar[ur]^{e_{XY}\ot\id} \ar[dr]_{\mu\ot\id} & & \bc_*(Z) \\ |
638 & CH_*(X, Z) \ot \bc_*(X) \ar[ur]_{e_{XZ}} & |
637 & CH_*(X, Z) \ot \bc_*(X) \ar[ur]_{e_{XZ}} & |
639 } \] |
638 } \] |
640 Here $\mu:CH_*(X, Y) \ot CH_*(Y, Z)\to CH_*(X, Z)$ is the map induced by composition |
639 Here $\mu:CH_*(X, Y) \ot CH_*(Y, Z)\to CH_*(X, Z)$ is the map induced by composition |
641 of homeomorphisms. |
640 of homeomorphisms. |
642 \end{prop} |
641 \end{thm} |
643 |
642 |
644 \begin{proof} |
643 \begin{proof} |
645 The strategy of the proof is similar to that of Proposition \ref{CHprop}. |
644 The strategy of the proof is similar to that of Theorem \ref{thm:CH}. |
646 We will identify a subcomplex |
645 We will identify a subcomplex |
647 \[ |
646 \[ |
648 G_* \sub CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X) |
647 G_* \sub CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X) |
649 \] |
648 \] |
650 where it is easy to see that the two sides of the diagram are homotopic, then |
649 where it is easy to see that the two sides of the diagram are homotopic, then |
654 By definition, $p\ot q\ot b\in G_*$ if there is a disjoint union of balls in $X$ which |
653 By definition, $p\ot q\ot b\in G_*$ if there is a disjoint union of balls in $X$ which |
655 contains $|p| \cup p\inv(|q|) \cup |b|$. |
654 contains $|p| \cup p\inv(|q|) \cup |b|$. |
656 (If $p:P\times X\to Y$, then $p\inv(|q|)$ means the union over all $x\in P$ of |
655 (If $p:P\times X\to Y$, then $p\inv(|q|)$ means the union over all $x\in P$ of |
657 $p(x, \cdot)\inv(|q|)$.) |
656 $p(x, \cdot)\inv(|q|)$.) |
658 |
657 |
659 As in the proof of Proposition \ref{CHprop}, we can construct a homotopy |
658 As in the proof of Theorem \ref{thm:CH}, we can construct a homotopy |
660 between the upper and lower maps restricted to $G_*$. |
659 between the upper and lower maps restricted to $G_*$. |
661 This uses the facts that the maps agree on $CH_0(X, Y) \ot CH_0(Y, Z) \ot \bc_*(X)$, |
660 This uses the facts that the maps agree on $CH_0(X, Y) \ot CH_0(Y, Z) \ot \bc_*(X)$, |
662 that they are compatible with gluing, and the contractibility of $\bc_*(X)$. |
661 that they are compatible with gluing, and the contractibility of $\bc_*(X)$. |
663 |
662 |
664 We can now apply Lemma \ref{extension_lemma_c}, using a series of increasingly fine covers, |
663 We can now apply Lemma \ref{extension_lemma_c}, using a series of increasingly fine covers, |