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%!TEX root = ../blob1.tex |
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\section{Introduction} |
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We construct the ``blob complex'' $\bc_*(M; \cC)$ associated to an $n$-manifold $M$ and a ``linear $n$-category with strong duality'' $\cC$. This blob complex provides a simultaneous generalisation of several well-understood constructions: |
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\begin{itemize} |
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\item The vector space $H_0(\bc_*(M; \cC))$ is isomorphic to the usual topological quantum field theory invariant of $M$ associated to $\cC$. (See \S \ref{sec:fields} \nn{more specific}.) |
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\item When $n=1$, $\cC$ is just an associative algebroid, and $\bc_*(S^1; \cC)$ is quasi-isomorphic to the Hochschild complex $\HC_*(\cC)$. (See \S \ref{sec:hochschild}.) |
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\item When $\cC$ is the polynomial algebra $k[t]$, thought of as an n-category (see \S \ref{sec:comm_alg}), we have |
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that $\bc_*(M; k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$, the singular chains |
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on the configurations space of unlabeled points in $M$. |
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%$$H_*(\bc_*(M; k[t])) = H^{\text{sing}}_*(\Delta^\infty(M), k).$$ |
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\end{itemize} |
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The blob complex definition is motivated by the desire for a derived analogue of the usual TQFT Hilbert space (replacing quotient of fields by local relations with some sort of resolution), |
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and for a generalization of Hochschild homology to higher $n$-categories. We would also like to be able to talk about $\CM{M}{T}$ when $T$ is an $n$-category rather than a manifold. The blob complex allows us to do all of these! More detailed motivations are described in \S \ref{sec:motivations}. |
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The blob complex has good formal properties, summarized in \S \ref{sec:properties}. These include an action of $\CH{M}$, |
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extending the usual $\Homeo(M)$ action on the TQFT space $H_0$ (see Property \ref{property:evaluation}) and a gluing formula allowing calculations by cutting manifolds into smaller parts (see Property \ref{property:gluing}). |
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We expect applications of the blob complex to contact topology and Khovanov homology but do not address these in this paper. See \S \ref{sec:future} for slightly more detail. |
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\subsubsection{Structure of the paper} |
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The three subsections of the introduction explain our motivations in defining the blob complex (see \S \ref{sec:motivations}), summarise the formal properties of the blob complex (see \S \ref{sec:properties}) and outline anticipated future directions and applications (see \S \ref{sec:future}). |
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The first part of the paper (sections \S \ref{sec:fields}---\S \ref{sec:evaluation}) gives the definition of the blob complex, and establishes some of its properties. There are many alternative definitions of $n$-categories, and part of our difficulty defining the blob complex is simply explaining what we mean by an ``$n$-category with strong duality'' as one of the inputs. At first we entirely avoid this problem by introducing the notion of a `system of fields', and define the blob complex associated to an $n$-manifold and an $n$-dimensional system of fields. We sketch the construction of a system of fields from a $1$-category or from a pivotal $2$-category. |
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Nevertheless, when we attempt to establish all of the observed properties of the blob complex, we find this situation unsatisfactory. Thus, in the second part of the paper (section \S \ref{sec:ncats}) we pause and give yet another definition of an $n$-category, or rather a definition of an $n$-category with strong duality. (It appears that removing the duality conditions from our definition would make it more complicated rather than less.) We call these ``topological $n$-categories'', to differentiate them from previous versions. Moreover, we find that we need analogous $A_\infty$ $n$-categories, and we define these as well following very similar axioms. |
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The basic idea is that each potential definition of an $n$-category makes a choice about the `shape' of morphisms. We try to be as lax as possible: a topological $n$-category associates a vector space to every $B$ homeomorphic to the $n$-ball. These vector spaces glue together associatively, and we require that there is an action of the homeomorphism group. |
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For an $A_\infty$ $n$-category, we associate a chain complex instead of a vector space to each such $B$ and ask that the action of homeomorphisms extends to a suitably defined action of the complex of singular chains of homeomorphisms. The axioms for an $A_\infty$ $n$-category are designed to capture two main examples: the blob complexes of $n$-balls labelled by a topological $n$-category, and the complex $\CM{-}{T}$ of maps to a fixed target space $T$. |
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In \S \ref{sec:ainfblob} we explain how to construct a system of fields from a topological $n$-category, and give an alternative definition of the blob complex for an $A_\infty$ $n$-category on an $n$-manifold. Using these definitions, we show how to use the blob complex to `resolve' any topological $n$-category as an $A_\infty$ $n$-category, and relate the first and second definitions of the blob complex. We use the blob complex for $A_\infty$ $n$-categories to establish important properties of the blob complex (in both variants), in particular the `gluing formula' of Property \ref{property:gluing} below. |
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\nn{KW: the previous two paragraphs seem a little awkward to me, but I don't presently have a good idea for fixing them.} |
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Finally, later sections address other topics. Section \S \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra, thought of as a topological $n$-category, in terms of the topology of $M$. Section \S \ref{sec:deligne} states (and in a later edition of this paper, hopefully proves) a higher dimensional generalisation of the Deligne conjecture (that the little discs operad acts on Hochschild cohomology) in terms of the blob complex. The appendixes prove technical results about $\CH{M}$, and make connections between our definitions of $n$-categories and familar definitions for $n=1$ and $n=2$, as well as relating the $n=1$ case of our $A_\infty$ $n$-categories with usual $A_\infty$ algebras. |
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\nn{some more things to cover in the intro} |
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\begin{itemize} |
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\item related: we are being unsophisticated from a homotopy theory point of |
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view and using chain complexes in many places where we could get by with spaces |
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\item ? one of the points we make (far) below is that there is not really much |
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difference between (a) systems of fields and local relations and (b) $n$-cats; |
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thus we tend to switch between talking in terms of one or the other |
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\end{itemize} |
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\medskip\hrule\medskip |
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\subsection{Motivations} |
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\label{sec:motivations} |
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We will briefly sketch our original motivation for defining the blob complex. |
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\nn{this is adapted from an old draft of the intro; it needs further modification |
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in order to better integrate it into the current intro.} |
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As a starting point, consider TQFTs constructed via fields and local relations. |
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(See Section \ref{sec:tqftsviafields} or \cite{kw:tqft}.) |
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This gives a satisfactory treatment for semisimple TQFTs |
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(i.e.\ TQFTs for which the cylinder 1-category associated to an |
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$n{-}1$-manifold $Y$ is semisimple for all $Y$). |
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For non-semi-simple TQFTs, this approach is less satisfactory. |
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Our main motivating example (though we will not develop it in this paper) |
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is the (decapitated) $4{+}1$-dimensional TQFT associated to Khovanov homology. |
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It associates a bigraded vector space $A_{Kh}(W^4, L)$ to a 4-manifold $W$ together |
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with a link $L \subset \bd W$. |
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The original Khovanov homology of a link in $S^3$ is recovered as $A_{Kh}(B^4, L)$. |
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How would we go about computing $A_{Kh}(W^4, L)$? |
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For $A_{Kh}(B^4, L)$, the main tool is the exact triangle (long exact sequence) |
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relating resolutions of a crossing. |
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Unfortunately, the exactness breaks if we glue $B^4$ to itself and attempt |
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to compute $A_{Kh}(S^1\times B^3, L)$. |
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According to the gluing theorem for TQFTs-via-fields, gluing along $B^3 \subset \bd B^4$ |
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corresponds to taking a coend (self tensor product) over the cylinder category |
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associated to $B^3$ (with appropriate boundary conditions). |
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The coend is not an exact functor, so the exactness of the triangle breaks. |
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The obvious solution to this problem is to replace the coend with its derived counterpart. |
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This presumably works fine for $S^1\times B^3$ (the answer being the Hochschild homology |
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of an appropriate bimodule), but for more complicated 4-manifolds this leaves much to be desired. |
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If we build our manifold up via a handle decomposition, the computation |
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would be a sequence of derived coends. |
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A different handle decomposition of the same manifold would yield a different |
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sequence of derived coends. |
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To show that our definition in terms of derived coends is well-defined, we |
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would need to show that the above two sequences of derived coends yield the same answer. |
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This is probably not easy to do. |
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Instead, we would prefer a definition for a derived version of $A_{Kh}(W^4, L)$ |
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which is manifestly invariant. |
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(That is, a definition that does not |
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involve choosing a decomposition of $W$. |
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After all, one of the virtues of our starting point --- TQFTs via field and local relations --- |
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is that it has just this sort of manifest invariance.) |
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The solution is to replace $A_{Kh}(W^4, L)$, which is a quotient |
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\[ |
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\text{linear combinations of fields} \;\big/\; \text{local relations} , |
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\] |
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with an appropriately free resolution (the ``blob complex") |
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\[ |
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\cdots\to \bc_2(W, L) \to \bc_1(W, L) \to \bc_0(W, L) . |
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\] |
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Here $\bc_0$ is linear combinations of fields on $W$, |
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$\bc_1$ is linear combinations of local relations on $W$, |
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$\bc_2$ is linear combinations of relations amongst relations on $W$, |
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and so on. |
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None of the above ideas depend on the details of the Khovanov homology example, |
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so we develop the general theory in this paper and postpone specific applications |
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to later papers. |
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\subsection{Formal properties} |
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\label{sec:properties} |
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We now summarize the results of the paper in 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. That is, |
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for a fixed $n$-dimensional system of fields $\cC$, the association |
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\begin{equation*} |
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X \mapsto \bc_*^{\cC}(X) |
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\end{equation*} |
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is a functor from $n$-manifolds and homeomorphisms between them to chain complexes and isomorphisms between them. |
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\end{property} |
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The blob complex is also functorial with respect to $\cC$, although we will not address this in detail here. |
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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 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 $X_\text{cut}$ contains $Y \sqcup Y^\text{op}$ as a codimension $0$-submanifold of its boundary, write $X_\text{glued} = X_\text{cut} \bigcup_{Y}\selfarrow$ for the manifold obtained by gluing together $Y$ and $Y^\text{op}$. 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 $X_\mathrm{cut} \to X_\mathrm{glued}$, there is |
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a natural map |
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\[ |
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\bc_*(X_\mathrm{cut}) \to \bc_*(X_\mathrm{glued}) |
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\] |
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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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\begin{property}[Contractibility] |
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\label{property:contractibility}% |
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\nn{this holds with field coefficients, or more generally when |
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the map to 0-th homology has a splitting; need to fix statement} |
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The blob complex on an $n$-ball is contractible in the sense that it is quasi-isomorphic to its $0$-th homology. Moreover, the $0$-th homology of balls can be canonically identified with the original $n$-category $\cC$. |
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\begin{equation} |
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\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)} |
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\end{equation} |
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\end{property} |
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\begin{property}[Skein modules] |
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\label{property: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 $\cC$. (See \S \ref{sec:local-relations}.) |
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\begin{equation*} |
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H_0(\bc_*^{\cC}(X)) \iso A^{\cC}(X) |
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\end{equation*} |
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\end{property} |
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\begin{property}[Hochschild homology when $X=S^1$] |
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\label{property: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_*^{\cC}(S^1) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & \HC_*(\cC).} |
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\end{equation*} |
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\end{property} |
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In the following $\CH{X}$ is the singular chain complex of the space of homeomorphisms of $X$, fixed on $\bdy X$. |
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\begin{property}[$C_*(\Homeo(-))$ action] |
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\label{property:evaluation}% |
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There is a chain map |
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\begin{equation*} |
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\ev_X: \CH{X} \tensor \bc_*(X) \to \bc_*(X). |
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\end{equation*} |
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Restricted to $C_0(\Homeo(X))$ this is just the action of homeomorphisms described in Property \ref{property:functoriality}. |
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\nn{should probably say something about associativity here (or not?)} |
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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. |
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\begin{equation*} |
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\xymatrix@C+2cm{ |
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\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) \\ |
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\CH{X} \otimes \bc_*(X) |
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\ar[r]_{\ev_{X}} \ar[u]^{\gl^{\Homeo}_Y \otimes \gl_Y} & |
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\bc_*(X) \ar[u]_{\gl_Y} |
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} |
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\end{equation*} |
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\nn{unique up to homotopy?} |
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\end{property} |
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Since the blob complex is functorial in the manifold $X$, we can use this to build a chain map |
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$$ev_{X \to Y} : \CH{X \to Y} \tensor \bc_*(X) \to \bc_*(Y)$$ |
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satisfying corresponding conditions. |
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In \S \ref{sec:ncats} we introduce the notion of topological $n$-categories, from which we can construct systems of fields, as well as the notion of an $A_\infty$ $n$-category. |
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\begin{property}[Blob complexes of (products with) balls form an $A_\infty$ $n$-category] |
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\label{property:blobs-ainfty} |
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Let $\cC$ be a topological $n$-category. Let $Y$ be an $n{-}k$-manifold. |
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Define $A_*(Y, \cC)$ on each $m$-ball $D$, for $0 \leq m < k$, to be the set $$A_*(Y,\cC)(D) = A^\cC(Y \times D)$$ and on $k$-balls $D$ to be the set $$A_*(Y, \cC)(D) = \bc_*(Y \times D, \cC).$$ (When $m=k$ the subsets with fixed boundary conditions form a chain complex.) These sets have the structure of an $A_\infty$ $k$-category. |
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\end{property} |
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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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\end{rem} |
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There is a version of the blob complex for $\cC$ an $A_\infty$ $n$-category |
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instead of a garden variety $n$-category; this is described in \S \ref{sec:ainfblob}. |
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\begin{property}[Product formula] |
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Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. Let $\cC$ be an $n$-category. |
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Let $A_*(Y,\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology (see Property \ref{property:blobs-ainfty}). |
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Then |
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\[ |
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\bc_*(Y\times W, \cC) \simeq \bc_*(W, A_*(Y,\cC)) . |
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\] |
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Note on the right hand side we have the version of the blob complex for $A_\infty$ $n$-categories. |
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\end{property} |
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It seems reasonable to expect a generalization describing an arbitrary fibre bundle. See in particular \S \ref{moddecss} for the framework for such a statement. |
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\begin{property}[Gluing formula] |
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\label{property:gluing}% |
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\mbox{}% <-- gets the indenting right |
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\begin{itemize} |
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\item For any $(n-1)$-manifold $Y$, the blob homology of $Y \times I$ is |
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naturally an $A_\infty$ category. % We'll write $\bc_*(Y)$ for $\bc_*(Y \times I)$ below. |
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\item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob homology of $X$ is naturally an |
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$A_\infty$ module for $\bc_*(Y \times I)$. |
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\item For any $n$-manifold $X_\text{glued} = X_\text{cut} \bigcup_Y \selfarrow$, the blob complex $\bc_*(X_\text{glued})$ is the $A_\infty$ self-tensor product of |
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$\bc_*(X_\text{cut})$ as an $\bc_*(Y \times I)$-bimodule. |
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\begin{equation*} |
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\bc_*(X_\text{glued}) \simeq \bc_*(X_\text{cut}) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y \times I)}} \selfarrow |
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\end{equation*} |
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\end{itemize} |
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\end{property} |
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Finally, we state two more properties, which we will not prove in this paper. |
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\begin{property}[Mapping spaces] |
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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{property} |
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This says that we can recover the (homotopic) space of maps to $T$ via blob homology from local data. |
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\begin{property}[Higher dimensional Deligne conjecture] |
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\label{property:deligne} |
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The singular chains of the $n$-dimensional fat graph operad act on blob cochains. |
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\end{property} |
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See \S \ref{sec:deligne} for an explanation of the terms appearing here. The proof will appear elsewhere. |
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Properties \ref{property:functoriality}, \ref{property:gluing-map} and \ref{property:skein-modules} will be immediate from the definition given in |
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\S \ref{sec:blob-definition}, and we'll recall them at the appropriate points there. \todo{Make sure this gets done.} |
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Properties \ref{property:disjoint-union} and \ref{property:contractibility} are established in \S \ref{sec:basic-properties}. |
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Property \ref{property:hochschild} is established in \S \ref{sec:hochschild}, Property \ref{property:evaluation} in \S \ref{sec:evaluation}, |
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and Property \ref{property:gluing} in \S \ref{sec:gluing}. |
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\nn{need to say where the remaining properties are proved.} |
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\subsection{Future directions} |
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\label{sec:future} |
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Throughout, we have resisted the temptation to work in the greatest generality possible (don't worry, it wasn't that hard). |
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In most of the places where we say ``set" or ``vector space", any symmetric monoidal category would do. |
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\nn{maybe make similar remark about chain complexes and $(\infty, 0)$-categories} |
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More could be said about finite characteristic (there appears in be $2$-torsion in $\bc_1(S^2, \cC)$ for any spherical $2$-category $\cC$, for example). Much more could be said about other types of manifolds, in particular oriented, $\operatorname{Spin}$ and $\operatorname{Pin}^{\pm}$ manifolds, where boundary issues become more complicated. (We'd recommend thinking about boundaries as germs, rather than just codimension $1$ manifolds.) We've also take the path of least resistance by considering $\operatorname{PL}$ manifolds; there may be some differences for topological manifolds and smooth manifolds. |
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Many results in Hochschild homology can be understood `topologically' via the blob complex. For example, we expect that the shuffle product on the Hochschild homology of a commutative algebra $A$ simply corresponds to the gluing operation on $\bc_*(S^1 \times [0,1], A)$, but haven't investigated the details. |
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Most importantly, however, \nn{applications!} \nn{$n=2$ cases, contact, Kh} |
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\subsection{Thanks and acknowledgements} |
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We'd like to thank David Ben-Zvi, Michael Freedman, Vaughan Jones, Justin Roberts, Chris Schommer-Pries, Peter Teichner \nn{probably lots more} for many interesting and useful conversations. During this work, Kevin Walker has been at Microsoft Station Q, and Scott Morrison has been at Microsoft Station Q and the Miller Institute for Basic Research at UC Berkeley. |
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\medskip\hrule\medskip |
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Still to do: |
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\begin{itemize} |
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\item say something about starting with semisimple n-cat (trivial?? not trivial?) |
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\item Mention somewhere \cite{MR1624157} ``Skein homology''; it's not directly related, but has similar motivations. |
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\end{itemize} |
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