# HG changeset patch
# User Kevin Walker <kevin@canyon23.net>
# Date 1319328452 21600
# Node ID db365e67adf6f984040f26ce98d4817fa6f2b2b2
# Parent  75c1e11d0f25361c0b6149c7d1efd57a928ad615
finished with plitting axiom stuff (except that now the module definition needs to be updated)

diff -r 75c1e11d0f25 -r db365e67adf6 blob to-do
--- a/blob to-do	Sat Oct 22 13:26:53 2011 -0600
+++ b/blob to-do	Sat Oct 22 18:07:32 2011 -0600
@@ -3,22 +3,9 @@
 
 * need to change module axioms to follow changes in n-cat axioms; search for and destroy all the "Homeo_\bd"'s, add a v-cone axiom
 
-* probably should go through and refer to new splitting axiom when we need to choose refinements etc.
-**** in the proof that gluing in dimension < n is injective
-
-* revisit splitting axiom for system of fields; check use of it in small blobs lemma
-
 * framings and duality -- work out what's going on! (alternatively, vague-ify current statement)
 
 
-(for reference:
-	* places splitting axiom is used:
-	** in the proof that gluing in dimension < n is injective
-	** in the proof that D(a) is acyclic
-	** in the small blobs lemma
-)
-
-
 ====== minor/optional ======
 
 * consider proving the gluing formula for higher codimension manifolds with
@@ -42,3 +29,16 @@
 * SCOTT will go through appendix C.2 and make it better (Schulman's example?)
 
 * SCOTT: review/proof-read recent KW changes, especially colimit section and n-cat axioms
+
+
+
+
+
+
+(for reference:
+	* places splitting axiom is used:
+	** in the proof that gluing in dimension < n is injective
+	** in the proof that D(a) is acyclic
+	** in the small blobs lemma
+)
+
diff -r 75c1e11d0f25 -r db365e67adf6 text/ncat.tex
--- a/text/ncat.tex	Sat Oct 22 13:26:53 2011 -0600
+++ b/text/ncat.tex	Sat Oct 22 18:07:32 2011 -0600
@@ -680,6 +680,7 @@
 We need one additional axiom.
 It says, roughly, that given a $k$-ball $X$, $k<n$, and $c\in \cC(X)$, there exist sufficiently many splittings of $c$.
 We use this axiom in the proofs of \ref{lem:d-a-acyclic} and \ref{lem:colim-injective}.
+The analogous axiom for systems of fields is used in the proof of \ref{small-blobs-b}.
 All of the examples of (disk-like) $n$-categories we consider in this paper satisfy the axiom, but
 nevertheless we feel that it is too strong.
 In the future we would like to see this provisional version of the axiom replaced by something less restrictive.
@@ -689,14 +690,16 @@
 \begin{axiom}[Splittings]
 \label{axiom:splittings}
 Let $c\in \cC_k(X)$, with $0\le k < n$.
-Let $X = \cup_i X_i$ be a splitting of $X$.
+Let $s = \{X_i\}$ be a splitting of X (so $X = \cup_i X_i$).
+Let $\Homeo_\bd(X)$ denote homeomorphisms of $X$ which restrict to the identity on $\bd X$.
 \begin{itemize}
-\item (Version 1) Then there exists an embedded cell complex $S_c \sub X$, called the string locus of $c$,
-such that if the splitting $\{X_i\}$ is transverse to $S_c$ then $c$ splits along $\{X_i\}$.
-\item (Version 2) Consider the set of homeomorphisms $g:X\to X$ such that $c$ splits along $\{g(X_i)\}$.
+\item (Alternative 1) Consider the set of homeomorphisms $g:X\to X$ such that $c$ splits along $g(s)$.
 Then this subset of $\Homeo(X)$ is open and dense.
+Furthermore, if $s$ restricts to a splitting $\bd s$ of $\bd X$, and if $\bd c$ splits along $\bd s$, then the
+intersection of the set of such homeomorphisms $g$ with $\Homeo_\bd(X)$ is open and dense in $\Homeo_\bd(X)$.
+\item (Alternative 2) Then there exists an embedded cell complex $S_c \sub X$, called the string locus of $c$,
+such that if the splitting $s$ is transverse to $S_c$ then $c$ splits along $s$.
 \end{itemize}
-\nn{same something about extension from boundary}
 \end{axiom}
 
 We note some consequences of Axiom \ref{axiom:splittings}.
@@ -827,6 +830,14 @@
 This constitutes the middle part ($P\times \{0\}$ above) of $\vcone(P)$.
 \end{proof}
 
+\begin{cor}
+For any $c\in \cC_k(X)$, the geometric realization of the poset of splittings of $c$ is contractible.
+\end{cor}
+
+\begin{proof}
+In the geometric realization, V-Cones become actual cones, allowing us to contract any cycle.
+\end{proof}
+
 
 \noop{ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
@@ -1753,6 +1764,7 @@
 \item $a_i$ is the image of some $a'_i\in \psi(x'_i)$, which in turn is the image
 of $b'_i$ and $b'_{i+1}$.
 \end{itemize}
+(This is possible by Axiom \ref{axiom:splittings}.)
 Now consider the diagrams
 \[ \xymatrix{
 	& \psi(x'_{i-1}) \ar[rd] & \\
diff -r 75c1e11d0f25 -r db365e67adf6 text/tqftreview.tex
--- a/text/tqftreview.tex	Sat Oct 22 13:26:53 2011 -0600
+++ b/text/tqftreview.tex	Sat Oct 22 18:07:32 2011 -0600
@@ -198,7 +198,7 @@
 Let $c\in \cC_k(X)$ and let $Y\sub X$ be a codimension 1 properly embedded submanifold of $X$.
 Then for most small perturbations of $Y$ (e.g.\ for an open dense
 subset of such perturbations, or for all perturbations satisfying
-a transversality condition, c.f. Axiom \ref{axiom:splittings} much later) $c$ splits along $Y$.
+a transversality condition; c.f. Axiom \ref{axiom:splittings} much later) $c$ splits along $Y$.
 (In Example \ref{ex:maps-to-a-space(fields)}, $c$ splits along all such $Y$.
 In Example \ref{ex:traditional-n-categories(fields)}, $c$ splits along $Y$ so long as $Y$ 
 is in general position with respect to the cell decomposition