# HG changeset patch
# User Kevin Walker <kevin@canyon23.net>
# Date 1273256319 25200
# Node ID ae5a542c958e2f6e08a4bde3f867d1e4fb71f693
# Parent  2a5d54f5180889c4fa9044929a1b735f3d515830
hochschild stuff

diff -r 2a5d54f51808 -r ae5a542c958e text/blobdef.tex
--- a/text/blobdef.tex	Wed May 05 22:58:45 2010 -0700
+++ b/text/blobdef.tex	Fri May 07 11:18:39 2010 -0700
@@ -135,6 +135,9 @@
 (The case $B_i = B_j$ is allowed.
 If $B_i \sub B_j$ the boundaries of $B_i$ and $B_j$ are allowed to intersect.)
 If a blob has no other blobs strictly contained in it, we call it a twig blob.
+\nn{need to allow the case where $B\to X$ is not an embedding
+on $\bd B$.  this is because any blob diagram on $X_{cut}$ should give rise to one on $X_{gl}$
+and blobs are allowed to meet $\bd X$.}
 \item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$.
 (These are implied by the data in the next bullets, so we usually
 suppress them from the notation.)
@@ -188,6 +191,11 @@
 The $(-1)^j$ factors imply that the terms of $\bd^2(b)$ all cancel.
 Thus we have a chain complex.
 
+We define the {\it support} of a blob diagram $b$, $\supp(b) \sub X$, 
+to be the union of the blobs of $b$.
+For $y \in \bc_*(X)$ with $y = \sum c_i b_i$ ($c_i$ a non-zero number, $b_i$ a blob diagram),
+we define $\supp(y) \deq \bigcup_i \supp(b_i)$.
+
 We note that blob diagrams in $X$ have a structure similar to that of a simplicial set,
 but with simplices replaced by a more general class of combinatorial shapes.
 Let $P$ be the minimal set of (isomorphisms classes of) polyhedra which is closed under products
diff -r 2a5d54f51808 -r ae5a542c958e text/hochschild.tex
--- a/text/hochschild.tex	Wed May 05 22:58:45 2010 -0700
+++ b/text/hochschild.tex	Fri May 07 11:18:39 2010 -0700
@@ -416,8 +416,7 @@
 and the inclusion $G''_* \sub G'_*$ is a homotopy equivalence.
 For $G'_*$ the proof is the same as in (\ref{bcontract}), except that the splitting
 $G'_0 \to H_0(G'_*)$ concentrates the point labels at two points to the right and left of $*$.
-For $G''_*$ we note that any cycle is supported \nn{need to establish terminology for this; maybe
-in ``basic properties" section above} away from $*$.
+For $G''_*$ we note that any cycle is supported away from $*$.
 Thus any cycle lies in the image of the normal blob complex of a disjoint union
 of two intervals, which is contractible by (\ref{bcontract}) and (\ref{disj-union-contract}).
 Finally, it is easy to see that the inclusion
@@ -448,13 +447,25 @@
 This completes the proof that $i: K''_* \to K'_*$ is a homotopy equivalence.
 \nn{need to say above more clearly and settle on notation/terminology}
 
-Finally, we show that $K''_*$ is contractible.
-\nn{need to also show that $H_0$ is the right thing; easy, but I won't do it now}
-Let $x$ be a cycle in $K''_*$.
+Finally, we show that $K''_*$ is contractible with $H_0\cong C$.
+This is similar to the proof of Proposition \ref{bcontract}, but a bit more
+complicated since there is no single blob which contains the support of all blob diagrams
+in $K''_*$.
+Let $x$ be a cycle of degree greater than zero in $K''_*$.
 The union of the supports of the diagrams in $x$ does not contain $*$, so there exists a
 ball $B \subset S^1$ containing the union of the supports and not containing $*$.
-Adding $B$ as a blob to $x$ gives a contraction.
-\nn{need to say something else in degree zero}
+Adding $B$ as an outermost blob to each summand of $x$ gives a chain $y$ with $\bd y = x$.
+Thus $H_i(K''_*) \cong 0$ for $i> 0$ and $K''_*$ is contractible.
+
+To see that $H_0(K''_*) \cong C$, consider the map $p: K''_0 \to C$ which sends a 0-blob
+diagram to the product of its labeled points.
+$p$ is clearly surjective.
+It's also easy to see that $p(\bd K''_1) = 0$.
+Finally, if $p(y) = 0$ then there exists a blob $B \sub S^1$ which contains
+all of the labeled points (other than *) of all of the summands of $y$.
+This allows us to construct $x\in K''_1$ such that $\bd x = y$.
+(The label of $B$ is the restriction of $y$ to $B$.)
+It follows that $H_0(K''_*) \cong C$.
 \end{proof}
 
 \medskip