--- a/text/evmap.tex Wed Oct 28 00:54:35 2009 +0000
+++ b/text/evmap.tex Wed Oct 28 02:44:29 2009 +0000
@@ -5,7 +5,7 @@
Let $CD_*(X, Y)$ denote $C_*(\Diff(X \to Y))$, the singular chain complex of
the space of diffeomorphisms
-\nn{or homeomorphisms}
+\nn{or homeomorphisms; need to fix the diff vs homeo inconsistency}
between the $n$-manifolds $X$ and $Y$ (extending a fixed diffeomorphism $\bd X \to \bd Y$).
For convenience, we will permit the singular cells generating $CD_*(X, Y)$ to be more general
than simplices --- they can be based on any linear polyhedron.
@@ -19,14 +19,22 @@
}
On $CD_0(X, Y) \otimes \bc_*(X)$ it agrees with the obvious action of $\Diff(X, Y)$ on $\bc_*(X)$
(Proposition (\ref{diff0prop})).
-For any splittings $X = X_1 \cup X_2$ and $Y = Y_1 \cup Y_2$,
+For any compatible splittings $X\to X\sgl$ and $Y\to Y\sgl$,
the following diagram commutes up to homotopy
\eq{ \xymatrix{
- CD_*(X, Y) \otimes \bc_*(X) \ar[r]^{e_{XY}} & \bc_*(Y) \\
- CD_*(X_1, Y_1) \otimes CD_*(X_2, Y_2) \otimes \bc_*(X_1) \otimes \bc_*(X_2)
- \ar@/_4ex/[r]_{e_{X_1Y_1} \otimes e_{X_2Y_2}} \ar[u]^{\gl \otimes \gl} &
- \bc_*(Y_1) \otimes \bc_*(Y_2) \ar[u]_{\gl}
+ CD_*(X\sgl, Y\sgl) \otimes \bc_*(X\sgl) \ar[r]^(.7){e_{X\sgl Y\sgl}} & \bc_*(Y\sgl) \\
+ CD_*(X, Y) \otimes \bc_*(X)
+ \ar@/_4ex/[r]_{e_{XY}} \ar[u]^{\gl \otimes \gl} &
+ \bc_*(Y) \ar[u]_{\gl}
} }
+%For any splittings $X = X_1 \cup X_2$ and $Y = Y_1 \cup Y_2$,
+%the following diagram commutes up to homotopy
+%\eq{ \xymatrix{
+% CD_*(X, Y) \otimes \bc_*(X) \ar[r]^{e_{XY}} & \bc_*(Y) \\
+% CD_*(X_1, Y_1) \otimes CD_*(X_2, Y_2) \otimes \bc_*(X_1) \otimes \bc_*(X_2)
+% \ar@/_4ex/[r]_{e_{X_1Y_1} \otimes e_{X_2Y_2}} \ar[u]^{\gl \otimes \gl} &
+% \bc_*(Y_1) \otimes \bc_*(Y_2) \ar[u]_{\gl}
+%} }
Any other map satisfying the above two properties is homotopic to $e_X$.
\end{prop}