--- a/text/basic_properties.tex Wed Oct 28 00:54:35 2009 +0000
+++ b/text/basic_properties.tex Wed Oct 28 02:44:29 2009 +0000
@@ -27,7 +27,7 @@
we have a splitting $s: H_0(\bc_*(B^n, c)) \to \bc_0(B^n; c)$
of the quotient map
$p: \bc_0(B^n; c) \to H_0(\bc_*(B^n, c))$.
-For example, this is always the case if you coefficient ring is a field.
+For example, this is always the case if the coefficient ring is a field.
Then
\begin{prop} \label{bcontract}
For all $c \in \cC(\bd B^n)$ the natural map $p: \bc_*(B^n, c) \to H_0(\bc_*(B^n, c))$
@@ -66,14 +66,14 @@
\begin{prop}
For fixed fields ($n$-cat), $\bc_*$ is a functor from the category
-of $n$-manifolds and diffeomorphisms to the category of chain complexes and
+of $n$-manifolds and homeomorphisms to the category of chain complexes and
(chain map) isomorphisms.
\qed
\end{prop}
In particular,
\begin{prop} \label{diff0prop}
-There is an action of $\Diff(X)$ on $\bc_*(X)$.
+There is an action of $\Homeo(X)$ on $\bc_*(X)$.
\qed
\end{prop}
@@ -106,16 +106,16 @@
The above map is very far from being an isomorphism, even on homology.
This will be fixed in Section \ref{sec:gluing} below.
-\nn{Next para not need, since we already use bullet = gluing notation above(?)}
+%\nn{Next para not needed, since we already use bullet = gluing notation above(?)}
-An instance of gluing we will encounter frequently below is where $X = X_1 \du X_2$
-and $X\sgl = X_1 \cup_Y X_2$.
-(Typically one of $X_1$ or $X_2$ is a disjoint union of balls.)
-For $x_i \in \bc_*(X_i)$, we introduce the notation
-\eq{
- x_1 \bullet x_2 \deq \gl(x_1 \otimes x_2) .
-}
-Note that we have resumed our habit of omitting boundary labels from the notation.
+%An instance of gluing we will encounter frequently below is where $X = X_1 \du X_2$
+%and $X\sgl = X_1 \cup_Y X_2$.
+%(Typically one of $X_1$ or $X_2$ is a disjoint union of balls.)
+%For $x_i \in \bc_*(X_i)$, we introduce the notation
+%\eq{
+% x_1 \bullet x_2 \deq \gl(x_1 \otimes x_2) .
+%}
+%Note that we have resumed our habit of omitting boundary labels from the notation.