
diff -r f4fc8028aacb -r 8174b33dda66 blob1.tex
--- a/blob1.tex	Sat Feb 09 01:01:03 2008 +0000
+++ b/blob1.tex	Sat Feb 09 15:16:43 2008 +0000
@@ -300,7 +300,7 @@
 \item A local relation field $u \in U(B; c)$
 (same $c$ as previous bullet).
 \end{itemize}
-%(Note that the the field $c$ is determined (implicitly) as the boundary of $u$ and/or $r$,
+%(Note that the field $c$ is determined (implicitly) as the boundary of $u$ and/or $r$,
 %so we will omit $c$ from the notation.)
 Define $\bc_1(X)$ to be the space of all finite linear combinations of
 1-blob diagrams, modulo the simple relations relating labels of 0-cells and
@@ -966,7 +966,7 @@
 $\eta: E \to R$, where $R$ is some $j$-dimensional polyhedron.
 The vertices of $R$ are associated to $k$-cells of the $K_\alpha$, and thence to points of $P$.
 If we triangulate $R$ (without introducing new vertices), we can linearly extend
-a map from the the vertices of $R$ into $P$ to a map of all of $R$ into $P$.
+a map from the vertices of $R$ into $P$ to a map of all of $R$ into $P$.
 Let $\cN$ be the set of all $\beta$ for which $K_\beta$ has a $k$-cell whose boundary meets
 the $k{-}j$-cell corresponding to $E$.
 For each $\beta \in \cN$, let $\{q_{\beta i}\}$ be the set of points in $P$ associated to the aforementioned $k$-cells.
@@ -989,7 +989,7 @@
 Since $u(0, p, x) = p$ for all $p\in P$ and $x\in X$, $F(0, p, x) = f(p, x)$ for all $p$ and $x$.
 Therefore $F$ is a homotopy from $f$ to something.
 
-Next we show that the the $K_\alpha$'s are sufficiently fine cell decompositions,
+Next we show that the $K_\alpha$'s are sufficiently fine cell decompositions,
 then $F$ is a homotopy through diffeomorphisms.
 We must show that the derivative $\pd{F}{x}(t, p, x)$ is non-singular for all $(t, p, x)$.
 We have




