author | Scott Morrison <scott@tqft.net> |
Mon, 25 Oct 2010 13:08:15 -0700 | |
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\usepackage{amssymb,amsfonts,amsmath,amsthm} |
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\input{preamble} |
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\input{../text/kw_macros} |
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\contributor{Submitted to Proceedings |
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of the National Academy of Sciences of the United States of America} |
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%\url{www.pnas.org/cgi/doi/10.1073/pnas.0709640104} |
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%% Molecular, Murcia, Spain}, \and Franklin Sonnery\affil{2}{}} |
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\author{Scott Morrison\affil{1}{Miller Institute for Basic Research, UC Berkeley, CA 94704, USA} \and Kevin Walker\affil{2}{Microsoft Station Q, 2243 CNSI Building, UC Santa Barbara, CA 93106, USA}} |
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\contributor{Submitted to Proceedings of the National Academy of Sciences |
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of the United States of America} |
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\section{} |
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\nn{ |
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background: TQFTs are important, historically, semisimple categories well-understood. |
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Many new examples arising recently which do not fit this framework, e.g. SW and OS theory. |
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These have more complicated gluing formulas (\cite{1003.0598,1005.1248}, etc); |
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it would be nice to give generalized TQFT axioms that encompass these. |
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Triangulated categories are important; often calculations are via exact sequences, |
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and the standard TQFT constructions are quotients, which destroy exactness. |
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A first attempt to deal with this might be to replace all the tensor products in gluing formulas |
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with derived tensor products (cite Kh?). |
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However, in this approach it's probably difficult to prove invariance of constructions, |
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because they depend on explicit presentations of the manifold. |
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We'll give a manifestly invariant construction, |
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and deduce gluing formulas based on derived (actually, $A_\infty$) tensor products.} |
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\section{Definitions} |
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\subsection{$n$-categories} |
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\nn{ |
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Axioms for $n$-categories, examples (maps, string diagrams) |
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} |
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\nn{ |
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Decide if we need a friendlier, skein-module version. |
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} |
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\subsection{The blob complex} |
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\subsubsection{Decompositions of manifolds} |
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\nn{Mention that the axioms for $n$-categories can be stated in terms of decompositions of balls} |
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\subsubsection{Homotopy colimits} |
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\nn{How can we extend an $n$-category from balls to arbitrary manifolds?} |
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\nn{In practice, this gives the old definition} |
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\subsubsection{} |
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\section{Properties of the blob complex} |
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\subsection{Formal properties} |
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\label{sec:properties} |
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The blob complex enjoys the following list of formal properties. |
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\begin{property}[Functoriality] |
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\label{property:functoriality}% |
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The blob complex is functorial with respect to homeomorphisms. |
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That is, |
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for a fixed $n$-dimensional system of fields $\cF$, the association |
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\begin{equation*} |
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X \mapsto \bc_*(X; \cF) |
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\end{equation*} |
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is a functor from $n$-manifolds and homeomorphisms between them to chain |
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complexes and isomorphisms between them. |
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\end{property} |
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As a consequence, there is an action of $\Homeo(X)$ on the chain complex $\bc_*(X; \cF)$; |
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this action is extended to all of $C_*(\Homeo(X))$ in Theorem \ref{thm:CH} below. |
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\begin{property}[Disjoint union] |
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\label{property:disjoint-union} |
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The blob complex of a disjoint union is naturally isomorphic to the tensor product of the blob complexes. |
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\begin{equation*} |
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\bc_*(X_1 \du X_2) \iso \bc_*(X_1) \tensor \bc_*(X_2) |
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\end{equation*} |
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\end{property} |
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If an $n$-manifold $X$ contains $Y \sqcup Y^\text{op}$ as a codimension $0$ submanifold of its boundary, |
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write $X_\text{gl} = X \bigcup_{Y}\selfarrow$ for the manifold obtained by gluing together $Y$ and $Y^\text{op}$. |
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Note that this includes the case of gluing two disjoint manifolds together. |
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\begin{property}[Gluing map] |
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\label{property:gluing-map}% |
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%If $X_1$ and $X_2$ are $n$-manifolds, with $Y$ a codimension $0$-submanifold of $\bdy X_1$, and $Y^{\text{op}}$ a codimension $0$-submanifold of $\bdy X_2$, there is a chain map |
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%\begin{equation*} |
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%\gl_Y: \bc_*(X_1) \tensor \bc_*(X_2) \to \bc_*(X_1 \cup_Y X_2). |
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%\end{equation*} |
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Given a gluing $X \to X_\mathrm{gl}$, there is |
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a natural map |
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\[ |
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\bc_*(X) \to \bc_*(X_\mathrm{gl}) |
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\] |
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(natural with respect to homeomorphisms, and also associative with respect to iterated gluings). |
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\end{property} |
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|
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\begin{property}[Contractibility] |
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\label{property:contractibility}% |
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With field coefficients, the blob complex on an $n$-ball is contractible in the sense |
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that it is homotopic to its $0$-th homology. |
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Moreover, the $0$-th homology of balls can be canonically identified with the vector spaces |
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associated by the system of fields $\cF$ to balls. |
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\begin{equation*} |
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\xymatrix{\bc_*(B^n;\cF) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*(B^n;\cF)) \ar[r]^(0.6)\iso & A_\cF(B^n)} |
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\end{equation*} |
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\end{property} |
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|
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\nn{Properties \ref{property:functoriality} will be immediate from the definition given in |
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\S \ref{sec:blob-definition}, and we'll recall it at the appropriate point there. |
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Properties \ref{property:disjoint-union}, \ref{property:gluing-map} and |
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\ref{property:contractibility} are established in \S \ref{sec:basic-properties}.} |
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|
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\subsection{Specializations} |
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\label{sec:specializations} |
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|
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The blob complex is a simultaneous generalization of the TQFT skein module construction and of Hochschild homology. |
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|
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\begin{thm}[Skein modules] |
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\label{thm:skein-modules} |
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The $0$-th blob homology of $X$ is the usual |
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(dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
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by $\cF$. |
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\begin{equation*} |
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H_0(\bc_*(X;\cF)) \iso A_{\cF}(X) |
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\end{equation*} |
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\end{thm} |
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\begin{thm}[Hochschild homology when $X=S^1$] |
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\label{thm:hochschild} |
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The blob complex for a $1$-category $\cC$ on the circle is |
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quasi-isomorphic to the Hochschild complex. |
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\begin{equation*} |
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\xymatrix{\bc_*(S^1;\cC) \ar[r]^(0.47){\iso}_(0.47){\text{qi}} & \HC_*(\cC).} |
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\end{equation*} |
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\end{thm} |
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|
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Proposition \ref{thm:skein-modules} is immediate from the definition, and |
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Theorem \ref{thm:hochschild} is established by extending the statement to bimodules as well as categories, then verifying that the universal properties of Hochschild homology also hold for $\bc_*(S^1; -)$. |
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|
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\subsection{Structure of the blob complex} |
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\label{sec:structure} |
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|
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In the following $\CH{X} = C_*(\Homeo(X))$ is the singular chain complex of the space of homeomorphisms of $X$, fixed on $\bdy X$. |
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\begin{thm}[$C_*(\Homeo(-))$ action] |
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\label{thm:CH}\label{thm:evaluation} |
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There is a chain map |
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\begin{equation*} |
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e_X: \CH{X} \tensor \bc_*(X) \to \bc_*(X). |
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\end{equation*} |
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such that |
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\begin{enumerate} |
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\item Restricted to $CH_0(X)$ this is the action of homeomorphisms described in Property \ref{property:functoriality}. |
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|
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\item For |
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any codimension $0$-submanifold $Y \sqcup Y^\text{op} \subset \bdy X$ the following diagram |
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(using the gluing maps described in Property \ref{property:gluing-map}) commutes (up to homotopy). |
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\begin{equation*} |
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\xymatrix@C+0.3cm{ |
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\CH{X} \otimes \bc_*(X) |
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\ar[r]_{e_{X}} \ar[d]^{\gl^{\Homeo}_Y \otimes \gl_Y} & |
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\bc_*(X) \ar[d]_{\gl_Y} \\ |
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\CH{X \bigcup_Y \selfarrow} \otimes \bc_*(X \bigcup_Y \selfarrow) \ar[r]_<<<<<<<{e_{(X \bigcup_Y \scalebox{0.5}{\selfarrow})}} & \bc_*(X \bigcup_Y \selfarrow) |
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} |
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\end{equation*} |
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\end{enumerate} |
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|
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Futher, this map is associative, in the sense that the following diagram commutes (up to homotopy). |
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\begin{equation*} |
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\xymatrix{ |
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\CH{X} \tensor \CH{X} \tensor \bc_*(X) \ar[r]^<<<<<{\id \tensor e_X} \ar[d]^{\compose \tensor \id} & \CH{X} \tensor \bc_*(X) \ar[d]^{e_X} \\ |
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\CH{X} \tensor \bc_*(X) \ar[r]^{e_X} & \bc_*(X) |
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} |
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\end{equation*} |
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\end{thm} |
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Since the blob complex is functorial in the manifold $X$, this is equivalent to having chain maps |
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$$ev_{X \to Y} : \CH{X \to Y} \tensor \bc_*(X) \to \bc_*(Y)$$ |
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for any homeomorphic pair $X$ and $Y$, |
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satisfying corresponding conditions. |
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|
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\begin{thm}[Blob complexes of products with balls form an $A_\infty$ $n$-category] |
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\label{thm:blobs-ainfty} |
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Let $\cC$ be a topological $n$-category. |
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Let $Y$ be an $n{-}k$-manifold. |
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There is an $A_\infty$ $k$-category $\bc_*(Y;\cC)$, defined on each $m$-ball $D$, for $0 \leq m < k$, |
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to be the set $$\bc_*(Y;\cC)(D) = \cC(Y \times D)$$ and on $k$-balls $D$ to be the set |
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$$\bc_*(Y;\cC)(D) = \bc_*(Y \times D; \cC).$$ |
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(When $m=k$ the subsets with fixed boundary conditions form a chain complex.) |
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These sets have the structure of an $A_\infty$ $k$-category, with compositions coming from the gluing map in |
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Property \ref{property:gluing-map} and with the action of families of homeomorphisms given in Theorem \ref{thm:evaluation}. |
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\end{thm} |
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\begin{rem} |
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Perhaps the most interesting case is when $Y$ is just a point; then we have a way of building an $A_\infty$ $n$-category from a topological $n$-category. |
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We think of this $A_\infty$ $n$-category as a free resolution. |
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\end{rem} |
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This result is described in more detail as Example 6.2.8 of \cite{1009.5025} |
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The definition is in fact simpler, almost tautological, and we use a different notation, $\cl{\cC}(M)$. The next theorem describes the blob complex for product manifolds, in terms of the $A_\infty$ blob complex of the $A_\infty$ $n$-categories constructed as in the previous example. |
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%The notation is intended to reflect the close parallel with the definition of the TQFT skein module via a colimit. |
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\newtheorem*{thm:product}{Theorem \ref{thm:product}} |
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\begin{thm}[Product formula] |
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\label{thm:product} |
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Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. |
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Let $\cC$ be an $n$-category. |
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Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology (see Example \ref{ex:blob-complexes-of-balls}). |
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Then |
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\[ |
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\bc_*(Y\times W; \cC) \simeq \cl{\bc_*(Y;\cC)}(W). |
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\] |
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\end{thm} |
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The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps |
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(see \cite[\S7.1]{1009.5025}). |
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|
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Fix a topological $n$-category $\cC$, which we'll omit from the notation. |
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Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ category. |
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|
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\begin{thm}[Gluing formula] |
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\label{thm:gluing} |
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\mbox{}% <-- gets the indenting right |
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\begin{itemize} |
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\item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob complex of $X$ is naturally an |
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$A_\infty$ module for $\bc_*(Y)$. |
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|
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\item For any $n$-manifold $X_\text{gl} = X\bigcup_Y \selfarrow$, the blob complex $\bc_*(X_\text{gl})$ is the $A_\infty$ self-tensor product of |
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$\bc_*(X)$ as an $\bc_*(Y)$-bimodule: |
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\begin{equation*} |
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\bc_*(X_\text{gl}) \simeq \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y)}} \selfarrow |
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\end{equation*} |
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\end{itemize} |
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\end{thm} |
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\nn{Theorem \ref{thm:product} is proved in \S \ref{ss:product-formula}, and Theorem \ref{thm:gluing} in \S \ref{sec:gluing}.} |
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\section{Applications} |
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\label{sec:applications} |
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Finally, we give two applications of the above machinery. |
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\begin{thm}[Mapping spaces] |
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\label{thm:map-recon} |
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Let $\pi^\infty_{\le n}(T)$ denote the $A_\infty$ $n$-category based on maps |
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$B^n \to T$. |
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(The case $n=1$ is the usual $A_\infty$-category of paths in $T$.) |
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Then |
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$$\bc_*(X; \pi^\infty_{\le n}(T)) \simeq \CM{X}{T}.$$ |
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\end{thm} |
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This says that we can recover (up to homotopy) the space of maps to $T$ via blob homology from local data. |
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Note that there is no restriction on the connectivity of $T$ as in \cite[Theorem 3.8.6]{0911.0018}. |
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\nn{The proof appears in \S \ref{sec:map-recon}.} |
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|
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\begin{thm}[Higher dimensional Deligne conjecture] |
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\label{thm:deligne} |
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The singular chains of the $n$-dimensional surgery cylinder operad act on blob cochains. |
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Since the little $n{+}1$-balls operad is a suboperad of the $n$-dimensional surgery cylinder operad, |
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this implies that the little $n{+}1$-balls operad acts on blob cochains of the $n$-ball. |
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\end{thm} |
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\nn{See \S \ref{sec:deligne} for a full explanation of the statement, and the proof.} |
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%%%% BIBTEX |
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\bibliographystyle{alpha} |
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%%%% non-BIBTEX |
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%\begin{thebibliography}{} |
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% |
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\end{article} |
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