--- a/text/evmap.tex Mon Jul 19 08:21:06 2010 -0700
+++ b/text/evmap.tex Mon Jul 19 08:42:24 2010 -0700
@@ -36,9 +36,13 @@
}
\end{equation*}
\end{enumerate}
-Up to (iterated) homotopy, there is a unique family $\{e_{XY}\}$ of chain maps
-satisfying the above two conditions.
+Moreover, for any $m \geq 0$, we can find a family of chain maps $\{e_{XY}\}$
+satisfying the above two conditions which is $m$-connected. In particular, this means that the choice of chain map above is unique up to homotopy.
\end{thm}
+\begin{rem}
+Note that the statement doesn't quite give uniqueness up to iterated homotopy. We fully expect that this should actually be the case, but haven't been able to prove this.
+\end{rem}
+
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.
@@ -345,7 +349,7 @@
\begin{proof}
-There exists $\lambda > 0$ such that for every subset $c$ of the blobs of $b$ $\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < \lambda$ .
+There exists $\lambda > 0$ such that for every subset $c$ of the blobs of $b$ the set $\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < \lambda$ .
(Here we are using the fact that the blobs are
piecewise smooth or piecewise-linear and that $\bd c$ is collared.)
We need to consider all such $c$ because all generators appearing in
@@ -582,9 +586,6 @@
these two maps agree up to $m$-th order homotopy.
More precisely, one can show that the subcomplex of maps containing the various
$e_{m+1}$ candidates is contained in the corresponding subcomplex for $e_m$.
-\nn{now should remark that we have not, in fact, produced a contractible set of maps,
-but we have come very close}
-\nn{better: change statement of thm}
\medskip