blob: comparison text/evmap.tex

equal deleted inserted replaced

-:c5a35886cd82
+:93ce0ba3d2d7
 We also will use the abbreviated notation $CH_*(X) \deq CH_*(X, X)$.
 (For convenience, we will permit the singular cells generating $CH_*(X, Y)$ to be more general
 than simplices --- they can be based on any linear polyhedron.
 \nn{be more restrictive here?  does more need to be said?})
-\begin{prop}  \label{CHprop}
+\begin{thm}  \label{thm:CH}
 For $n$-manifolds $X$ and $Y$ there is a chain map
 \eq{
 e_{XY} : CH_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y)
 }
 such that
 \begin{enumerate}
 \item on $CH_0(X, Y) \otimes \bc_*(X)$ it agrees with the obvious action of
-$\Homeo(X, Y)$ on $\bc_*(X)$ (Property (\ref{property:functoriality})), and
+$\Homeo(X, Y)$ on $\bc_*(X)$  described in Property (\ref{property:functoriality}), and
 \item for any compatible splittings $X\to X\sgl$ and $Y\to Y\sgl$,
 the following diagram commutes up to homotopy
 \begin{equation*}
 \xymatrix@C+2cm{
 CH_*(X, Y) \otimes \bc_*(X)
 }
 \end{equation*}
 \end{enumerate}
 Up to (iterated) homotopy, there is a unique family $\{e_{XY}\}$ of chain maps
 satisfying the above two conditions.
-\end{prop}
+\end{thm}
 Before giving the proof, we state the essential technical tool of Lemma \ref{extension_lemma},
 and then give an outline of the method of proof.
 Without loss of generality, we will assume $X = Y$.
 The proof will be given in \S\ref{sec:localising}.
 \medskip
-Before diving into the details, we outline our strategy for the proof of Proposition \ref{CHprop}.
+Before diving into the details, we outline our strategy for the proof of Theorem \ref{thm:CH}.
 Let $p$ be a singular cell in $CH_k(X)$ and $b$ be a blob diagram in $\bc_*(X)$.
 We say that $p\ot b$ is {\it localizable} if there exists $V \sub X$ such that
 \begin{itemize}
 \item $V$ is homeomorphic to a disjoint union of balls, and
 \item $\supp(p) \cup \supp(b) \sub V$.
 Because we want not mere homotopy uniqueness but iterated homotopy uniqueness,
 we will similarly require that $W$ be contained in a slightly larger metric neighborhood of
 $\supp(p)\cup\supp(b)$, and so on.
-\medskip
+\begin{proof}[Proof of Theorem \ref{thm:CH}.]
-\begin{proof}[Proof of Proposition \ref{CHprop}.]
 We'll use the notation $|b| = \supp(b)$ and $|p| = \supp(p)$.
 Choose a metric on $X$.
 Choose a monotone decreasing sequence of positive real numbers $\ep_i$ converging monotonically to zero
 (e.g.\ $\ep_i = 2^{-i}$).
 The gluing map $X\sgl\to X$ induces a map
 \[
 	\gl:  R_*\ot CH_*(X, X) \otimes \bc_*(X)  \to R_*\ot CH_*(X\sgl, X \sgl) \otimes \bc_*(X \sgl) ,
 \]
 and it is easy to see that $\gl(G^m_*)\sub \ol{G}^m_*$.
-From this it follows that the diagram in the statement of Proposition \ref{CHprop} commutes.
+From this it follows that the diagram in the statement of Theorem \ref{thm:CH} commutes.
 \todo{this paragraph isn't very convincing, or at least I don't see what's going on}
 Finally we show that the action maps defined above are independent of
 the choice of metric (up to iterated homotopy).
 The arguments are very similar to ones given above, so we only sketch them.
 Using acyclic models, we can construct a homotopy from $e$ to $e'$ on $F_*$.
 We now observe that $CH_*(X, X) \ot \bc_*(X)$ retracts to $F_*$.
 Similar arguments show that this homotopy from $e$ to $e'$ is well-defined
 up to second order homotopy, and so on.
-This completes the proof of Proposition \ref{CHprop}.
+This completes the proof of Theorem \ref{thm:CH}.
 \end{proof}
 \begin{rem*}
 \label{rem:for-small-blobs}
 of $p(t,|b|)$ with some small balls.
 (Here ``small" is in terms of the metric on $X$ that we chose to construct $e$.)
 \end{rem*}
-\begin{prop}
+\begin{thm}
+\label{thm:CH-associativity}
 The $CH_*(X, Y)$ actions defined above are associative.
 That is, the following diagram commutes up to homotopy:
 \[ \xymatrix{
 & CH_*(Y, Z) \ot \bc_*(Y) \ar[dr]^{e_{YZ}} & \\
 CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X) \ar[ur]^{e_{XY}\ot\id} \ar[dr]_{\mu\ot\id} & & \bc_*(Z) \\
 & CH_*(X, Z) \ot \bc_*(X) \ar[ur]_{e_{XZ}} &
 } \]
 Here $\mu:CH_*(X, Y) \ot CH_*(Y, Z)\to CH_*(X, Z)$ is the map induced by composition
 of homeomorphisms.
-\end{prop}
+\end{thm}
 \begin{proof}
-The strategy of the proof is similar to that of Proposition \ref{CHprop}.
+The strategy of the proof is similar to that of Theorem \ref{thm:CH}.
 We will identify a subcomplex
 \[
 	G_* \sub CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X)
 \]
 where it is easy to see that the two sides of the diagram are homotopic, then
 By definition, $p\ot q\ot b\in G_*$ if there is a disjoint union of balls in $X$ which
 contains $|p| \cup p\inv(|q|) \cup |b|$.
 (If $p:P\times X\to Y$, then $p\inv(|q|)$ means the union over all $x\in P$ of
 $p(x, \cdot)\inv(|q|)$.)
-As in the proof of Proposition \ref{CHprop}, we can construct a homotopy
+As in the proof of Theorem \ref{thm:CH}, we can construct a homotopy
 between the upper and lower maps restricted to $G_*$.
 This uses the facts that the maps agree on $CH_0(X, Y) \ot CH_0(Y, Z) \ot \bc_*(X)$,
 that they are compatible with gluing, and the contractibility of $\bc_*(X)$.
 We can now apply Lemma \ref{extension_lemma_c}, using a series of increasingly fine covers,

changeset 437	93ce0ba3d2d7
parent 430	c5a35886cd82
child 438	0d62ea7c653d