# HG changeset patch # User Kevin Walker # Date 1297054423 28800 # Node ID ef503460486d513abd9c62d738411bd630b96b53 # Parent 2313b05f4906341303051191170da83acf7f3ef0 Edits from Aaron Mazel-Gee diff -r 2313b05f4906 -r ef503460486d text/basic_properties.tex --- a/text/basic_properties.tex Tue Jan 25 14:57:07 2011 -0800 +++ b/text/basic_properties.tex Sun Feb 06 20:53:43 2011 -0800 @@ -31,16 +31,16 @@ conditions to the notation. Suppose that for all $c \in \cC(\bd B^n)$ -we have a splitting $s: H_0(\bc_*(B^n, c)) \to \bc_0(B^n; c)$ +we have a splitting $s: H_0(\bc_*(B^n; c)) \to \bc_0(B^n; c)$ of the quotient map -$p: \bc_0(B^n; c) \to H_0(\bc_*(B^n, c))$. +$p: \bc_0(B^n; c) \to H_0(\bc_*(B^n; c))$. 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 $p: \bc_*(B^n, c) \to H_0(\bc_*(B^n, c))$ +For all $c \in \cC(\bd B^n)$ the natural map $p: \bc_*(B^n; c) \to H_0(\bc_*(B^n; c))$ is a chain homotopy equivalence -with inverse $s: H_0(\bc_*(B^n, c)) \to \bc_*(B^n; c)$. -Here we think of $H_0(\bc_*(B^n, c))$ as a 1-step complex concentrated in degree 0. +with inverse $s: H_0(\bc_*(B^n; c)) \to \bc_*(B^n; c)$. +Here we think of $H_0(\bc_*(B^n; c))$ as a 1-step complex concentrated in degree 0. \end{prop} \begin{proof} By assumption $p\circ s = \id$, so all that remains is to find a degree 1 map @@ -67,8 +67,13 @@ This follows from Properties \ref{property:disjoint-union} and \ref{property:contractibility}. \end{proof} -Recall the definition of the support of a blob diagram as the union of all the -blobs of the diagram. +%Recall the definition of the support of a blob diagram as the union of all the +%blobs of the diagram. +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 $\supp(y) \deq \bigcup_i \supp(b_i)$. + For future use we prove the following lemma. \begin{lemma} \label{support-shrink} diff -r 2313b05f4906 -r ef503460486d text/blobdef.tex --- a/text/blobdef.tex Tue Jan 25 14:57:07 2011 -0800 +++ b/text/blobdef.tex Sun Feb 06 20:53:43 2011 -0800 @@ -33,9 +33,11 @@ to define fields on these pieces. We of course define $\bc_0(X) = \cF(X)$. -(If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \cF(X; c)$ for each $c \in \cF(\bdy X)$. +In other words, $\bc_0(X)$ is just the vector space of all fields on $X$. + +(If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \cF(X; c)$ for $c \in \cF(\bdy X)$. +The blob complex $\bc_*(X; c)$ will depend on a fixed boundary condition $c\in \cF(\bdy X)$. We'll omit such boundary conditions from the notation in the rest of this section.) -In other words, $\bc_0(X)$ is just the vector space of all fields on $X$. We want the vector space $\bc_1(X)$ to capture ``the space of all local relations that can be imposed on $\bc_0(X)$". @@ -148,8 +150,8 @@ \item For any (possibly empty) configuration of blobs on an $n$-ball $D$, we can add $D$ itself as an outermost blob. (This is used in the proof of Proposition \ref{bcontract}.) -\item If $X'$ is obtained from $X$ by gluing, then any permissible configuration of blobs -on $X$ gives rise to a permissible configuration on $X'$. +\item If $X\sgl$ is obtained from $X$ by gluing, then any permissible configuration of blobs +on $X$ gives rise to a permissible configuration on $X\sgl$. (This is necessary for Proposition \ref{blob-gluing}.) \end{itemize} Combining these two operations can give rise to configurations of blobs whose complement in $X$ is not @@ -166,8 +168,8 @@ \end{align*} Here $A \cup B = [0,1] \times [-1,1] \times [0,1]$ and $C \cup D = [-1,0] \times [-1,1] \times [0,1]$. Now, $\{A\}$ is a valid configuration of blobs in $A \cup B$, -and $\{C\}$ is a valid configuration of blobs in $C \cup D$, -so we must allow $\{A, C\}$ as a configuration of blobs in $[-1,1]^2 \times [0,1]$. +and $\{D\}$ is a valid configuration of blobs in $C \cup D$, +so we must allow $\{A, D\}$ as a configuration of blobs in $[-1,1]^2 \times [0,1]$. Note however that the complement is not a manifold. \end{example} @@ -244,7 +246,7 @@ \label{defn:blobs} The $k$-th vector space $\bc_k(X)$ of the \emph{blob complex} of $X$ is the direct sum over all configurations of $k$ blobs in $X$ of the vector space of $k$-blob diagrams with that configuration, -modulo identifying the vector spaces for configurations that only differ by a permutation of the balls +modulo identifying the vector spaces for configurations that only differ by a permutation of the blobs by the sign of that permutation. The differential $\bc_k(X) \to \bc_{k-1}(X)$ is, as above, the signed sum of ways of forgetting one blob from the configuration, preserving the field $r$: @@ -263,11 +265,6 @@ is immediately obvious from the definition. A homeomorphism acts in an obvious way on blobs and on fields. -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 $\supp(y) \deq \bigcup_i \supp(b_i)$. - \begin{remark} \label{blobsset-remark} \rm 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. diff -r 2313b05f4906 -r ef503460486d text/tqftreview.tex --- a/text/tqftreview.tex Tue Jan 25 14:57:07 2011 -0800 +++ b/text/tqftreview.tex Sun Feb 06 20:53:43 2011 -0800 @@ -456,10 +456,10 @@ %$\bc_0(X) = \lf(X)$. \begin{defn} \label{defn:TQFT-invariant} -The TQFT invariant of $X$ associated to a system of fields $\cF$ and local relations $\cU$ is +The TQFT invariant of $X$ associated to a system of fields $\cC$ and local relations $U$ is $$A(X) \deq \lf(X) / U(X),$$ -where $\cU(X) \sub \lf(X)$ is the space of local relations in $\lf(X)$: -$\cU(X)$ is generated by fields of the form $u\bullet r$, where +where $U(X) \sub \lf(X)$ is the space of local relations in $\lf(X)$: +$U(X)$ is generated by fields of the form $u\bullet r$, where $u\in U(B)$ for some embedded $n$-ball $B\sub X$ and $r\in \cC(X\setmin B)$. \end{defn} The blob complex, defined in the next section,