# HG changeset patch # User Kevin Walker # Date 1282623595 25200 # Node ID 9e44c14699187d3d55a1e41ac3c2a92191f5c213 # Parent bb696f417f22b46a9669e43c17880c6221684036 more on small blobs diff -r bb696f417f22 -r 9e44c1469918 text/blobdef.tex --- a/text/blobdef.tex Sun Aug 22 21:10:39 2010 -0700 +++ b/text/blobdef.tex Mon Aug 23 21:19:55 2010 -0700 @@ -174,7 +174,12 @@ \end{defn} Given a gluing decomposition $M_0 \to M_1 \to \cdots \to M_m = X$, we say that a field is splittable along it if it is the image of a field on $M_0$. -In the example above, note that $$A \sqcup B \sqcup C \sqcup D \to (A \cup B) \sqcup (C \cup D) \to A \cup B \cup C \cup D$$ is a ball decomposition, but other sequences of gluings starting from $A \sqcup B \sqcup C \sqcup D$have intermediate steps which are not manifolds. +In the example above, note that +\[ + A \sqcup B \sqcup C \sqcup D \to (A \cup B) \sqcup (C \cup D) \to A \cup B \cup C \cup D +\] +is a ball decomposition, but other sequences of gluings starting from $A \sqcup B \sqcup C \sqcup D$ +have intermediate steps which are not manifolds. We'll now slightly restrict the possible configurations of blobs. %%%%% oops -- I missed the similar discussion after the definition diff -r bb696f417f22 -r 9e44c1469918 text/evmap.tex --- a/text/evmap.tex Sun Aug 22 21:10:39 2010 -0700 +++ b/text/evmap.tex Mon Aug 23 21:19:55 2010 -0700 @@ -56,6 +56,8 @@ If $b$ is a blob diagram in $\bc_*(X)$, define the {\it support} of $b$, denoted $\supp(b)$ or $|b|$, to be the union of the blobs of $b$. +For a general $k-chain$ $a\in \bc_k(X)$, define the support of $a$ to be the union +of the supports of the blob diagrams which appear in it. If $f: P\times X\to X$ is a family of homeomorphisms and $Y\sub X$, we say that $f$ is {\it supported on $Y$} if $f(p, x) = f(p', x)$ for all $x\in X\setmin Y$ and all $p,p'\in P$. @@ -63,6 +65,10 @@ again denoted $\supp(f)$ or $|f|$, to mean some particular choice of $Y$ such that $f$ is supported on $Y$. +If $f: M \cup (Y\times I) \to M$ is a collaring homeomorphism +(cf. end of \S \ref{ss:syst-o-fields}), +we say that $f$ is supported on $S\sub M$ if $f(x) = x$ for all $x\in M\setmin S$. + Fix $\cU$, an open cover of $X$. Define the ``small blob complex" $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(X)$ of all blob diagrams in which every blob is contained in some open set of $\cU$, @@ -81,11 +87,44 @@ For simplicity we will assume that all fields are splittable into small pieces, so that $\sbc_0(X) = \bc_0$. +(This is true for all of the examples presented in this paper.) Accordingly, we define $h_0 = 0$. +Next we define $h_1$. Let $b\in C_1$ be a 1-blob diagram. +Let $B$ be the blob of $b$. +We will construct a 1-chain $s(b)\in \sbc_1$ such that $\bd(s(b)) = \bd b$ +and the support of $s(b)$ is contained in $B$. +(If $B$ is not embedded in $X$, then we implicitly work in some term of a decomposition +of $X$ where $B$ is embedded. +See \ref{defn:configuration} and preceding discussion.) +It then follows from \ref{disj-union-contract} that we can choose +$h_1(b) \in \bc_1(X)$ such that $\bd(h_1(b)) = s(b) - b$. + +Roughly speaking, $s(b)$ consists of a series of 1-blob diagrams implementing a series +of small collar maps, plus a shrunken version of $b$. +The composition of all the collar maps shrinks $B$ to a sufficiently small ball. + Let $\cV_1$ be an auxiliary open cover of $X$, satisfying conditions specified below. -Let $B$ be the blob of $b$. +Let $b = (B, u, r)$, $u = \sum a_i$ be the label of $B$, $a_i\in \bc_0(B)$. +Choose a series of collar maps $f_j:\bc_0(B)\to\bc_0(B)$ such that each has support +contained in an open set of $\cV_1$ and the composition of the corresponding collar homeomorphisms +yields an embedding $g:B\to B$ such that $g(B)$ is contained in an open set of $\cV_1$. +\nn{need to say this better; maybe give fig} +Let $g_j:B\to B$ be the embedding at the $j$-th stage. +There are 1-blob diagrams $c_{ij} \in \bc_1(B)$ such that $c_{ij}$ is compatible with $\cV_1$ +and $\bd c_{ij} = g_j(a_i) = g_{j-1}(a_i)$. +Define +\[ + s(b) = \sum_{i,j} c_{ij} + g(b) +\] +and choose $h_1(b) \in \bc_1(X)$ such that +\[ + \bd(h_1(b)) = s(b) - b . +\] + +Next we define $h_2$. + \nn{...} @@ -94,13 +133,6 @@ - -%Let $k$ be the top dimension of $C_*$. -%The construction of $h$ will involve choosing various - - - - \end{proof} diff -r bb696f417f22 -r 9e44c1469918 text/tqftreview.tex --- a/text/tqftreview.tex Sun Aug 22 21:10:39 2010 -0700 +++ b/text/tqftreview.tex Mon Aug 23 21:19:55 2010 -0700 @@ -36,6 +36,7 @@ \subsection{Systems of fields} +\label{ss:syst-o-fields} Let $\cM_k$ denote the category with objects unoriented PL manifolds of dimension