blob: comparison text/intro.tex

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 \label{property:functoriality}%
 The blob complex is functorial with respect to homeomorphisms.
 That is,
 for a fixed $n$-dimensional system of fields $\cC$, the association
 \begin{equation*}
-X \mapsto \bc_*^{\cC}(X)
+X \mapsto \bc_*(X; \cC)
 \end{equation*}
 is a functor from $n$-manifolds and homeomorphisms between them to chain
 complexes and isomorphisms between them.
 \end{property}
-As a consequence, there is an action of $\Homeo(X)$ on the chain complex $\bc_*^\cC(X)$;
+As a consequence, there is an action of $\Homeo(X)$ on the chain complex $\bc_*(X; \cC)$;
 this action is extended to all of $C_*(\Homeo(X))$ in Theorem \ref{thm:evaluation} below.
 The blob complex is also functorial (indeed, exact) with respect to $\cC$,
 although we will not address this in detail here.
 \begin{property}[Contractibility]
 \label{property:contractibility}%
 With field coefficients, the blob complex on an $n$-ball is contractible in the sense that it is homotopic to its $0$-th homology.
 Moreover, the $0$-th homology of balls can be canonically identified with the vector spaces associated by the system of fields $\cC$ to balls.
 \begin{equation*}
-\xymatrix{\bc_*^{\cC}(B^n) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*^{\cC}(B^n)) \ar[r]^(0.6)\iso & \cC(B^n)}
+\xymatrix{\bc_*(B^n;\cC) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*(B^n;\cC)) \ar[r]^(0.6)\iso & \cC(B^n)}
 \end{equation*}
 \end{property}
 Properties \ref{property:functoriality} will be immediate from the definition given in
 \S \ref{sec:blob-definition}, and we'll recall it at the appropriate point there.
 The $0$-th blob homology of $X$ is the usual
 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$
 by $\cC$.
 (See \S \ref{sec:local-relations}.)
 \begin{equation*}
-H_0(\bc_*^{\cC}(X)) \iso A^{\cC}(X)
+H_0(\bc_*(X;\cC)) \iso A_{\cC}(X)
 \end{equation*}
 \end{thm:skein-modules}
 \newtheorem*{thm:hochschild}{Theorem \ref{thm:hochschild}}
 \begin{thm:hochschild}[Hochschild homology when $X=S^1$]
 The blob complex for a $1$-category $\cC$ on the circle is
 quasi-isomorphic to the Hochschild complex.
 \begin{equation*}
-\xymatrix{\bc_*^{\cC}(S^1) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & \HC_*(\cC).}
+\xymatrix{\bc_*(S^1;\cC) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & \HC_*(\cC).}
 \end{equation*}
 \end{thm:hochschild}
 Theorem \ref{thm:skein-modules} is immediate from the definition, and
 Theorem \ref{thm:hochschild} is established in \S \ref{sec:hochschild}.
 In the following $\CH{X}$ is the singular chain complex of the space of homeomorphisms of $X$, fixed on $\bdy X$.
 \newtheorem*{thm:CH}{Theorem \ref{thm:CH}}
-\begin{thm:CH}[$C_*(\Homeo(-))$ action]\mbox{}\\
+\begin{thm:CH}[$C_*(\Homeo(-))$ action]
-\vspace{-0.5cm}
 \label{thm:evaluation}%
 There is a chain map
 \begin{equation*}
 \ev_X: \CH{X} \tensor \bc_*(X) \to \bc_*(X).
 \end{equation*}
 \item For
 any codimension $0$-submanifold $Y \sqcup Y^\text{op} \subset \bdy X$ the following diagram
 (using the gluing maps described in Property \ref{property:gluing-map}) commutes (up to homotopy).
 \begin{equation*}
 \xymatrix@C+2cm{
-\CH{X \bigcup_Y \selfarrow} \otimes \bc_*(X \bigcup_Y \selfarrow) \ar[r]^<<<<<<<<<<<<{\ev_{(X \bigcup_Y \scalebox{0.5}{\selfarrow})}}    & \bc_*(X \bigcup_Y \selfarrow) \\
 \CH{X} \otimes \bc_*(X)
-\ar[r]_{\ev_{X}}  \ar[u]^{\gl^{\Homeo}_Y \otimes \gl_Y}  &
+\ar[r]_{\ev_{X}}  \ar[d]^{\gl^{\Homeo}_Y \otimes \gl_Y}  &
-\bc_*(X) \ar[u]_{\gl_Y}
+\bc_*(X) \ar[d]_{\gl_Y} \\
+\CH{X \bigcup_Y \selfarrow} \otimes \bc_*(X \bigcup_Y \selfarrow) \ar[r]^<<<<<<<<<<<<{\ev_{(X \bigcup_Y \scalebox{0.5}{\selfarrow})}}    & \bc_*(X \bigcup_Y \selfarrow)
 }
 \end{equation*}
 \end{enumerate}
 Moreover any such chain map is unique, up to an iterated homotopy.
 (That is, any pair of homotopies have a homotopy between them, and so on.)
 \newtheorem*{thm:CH-associativity}{Theorem \ref{thm:CH-associativity}}
 Further,
 \begin{thm:CH-associativity}
-\item The chain map of Theorem \ref{thm:CH} is associative, in the sense that the following diagram commutes (up to homotopy).
+The chain map of Theorem \ref{thm:CH} is associative, in the sense that the following diagram commutes (up to homotopy).
 \begin{equation*}
 \xymatrix{
 \CH{X} \tensor \CH{X} \tensor \bc_*(X) \ar[r]^<<<<<{\id \tensor \ev_X} \ar[d]^{\compose \tensor \id} & \CH{X} \tensor \bc_*(X) \ar[d]^{\ev_X} \\
 \CH{X} \tensor \bc_*(X) \ar[r]^{\ev_X} & \bc_*(X)
 }

changeset 454	3377d4db80d9
parent 437	93ce0ba3d2d7
child 464	6c760675d461