author | Kevin Walker <kevin@canyon23.net> |
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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'' $\cC$. |
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This blob complex provides a simultaneous generalization 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 $\cC$. |
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(See Theorem \ref{thm:skein-modules} later in the introduction and \S \ref{sec:constructing-a-tqft}.) |
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\item When $n=1$ and $\cC$ is just a 1-category (e.g.\ an associative algebra), |
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the blob complex $\bc_*(S^1; \cC)$ is quasi-isomorphic to the Hochschild complex $\HC_*(\cC)$. |
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(See Theorem \ref{thm:hochschild} and \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 configuration 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 |
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(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. |
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We would also like to be able to talk about $\CM{M}{T}$ when $T$ is an $n$-category rather than a manifold. |
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The blob complex gives us 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}. |
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These include an action of $\CH{M}$, |
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extending the usual $\Homeo(M)$ action on the TQFT space $H_0$ (see Theorem \ref{thm:evaluation}) and a gluing |
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formula allowing calculations by cutting manifolds into smaller parts (see Theorem \ref{thm: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. |
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See \S \ref{sec:future} for slightly more detail. |
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\subsubsection{Structure of the paper} |
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The subsections of the introduction explain our motivations in defining the blob complex (see \S \ref{sec:motivations}), |
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summarise the formal properties of the blob complex (see \S \ref{sec:properties}), describe known specializations (see \S \ref{sec:specializations}), outline the major results of the paper (see \S \ref{sec:structure} and \S \ref{sec:applications}) |
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and outline anticipated future directions (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, |
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and establishes some of its properties. |
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There are many alternative definitions of $n$-categories, and part of our difficulty defining the blob complex is |
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simply explaining what we mean by an ``$n$-category with strong duality'' as one of the inputs. |
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At first we entirely avoid this problem by introducing the notion of a `system of fields', and define the blob complex |
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associated to an $n$-manifold and an $n$-dimensional system of fields. |
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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, |
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we find this situation unsatisfactory. |
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Thus, in the second part of the paper (\S\S \ref{sec:ncats}-\ref{sec:ainfblob}) we give yet another |
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definition of an $n$-category, or rather a definition of an $n$-category with strong duality. |
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(It appears that removing the duality conditions from our definition would make it more complicated rather than less.) |
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We call these ``topological $n$-categories'', to differentiate them from previous versions. |
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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. |
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We try to be as lax as possible: a topological $n$-category associates a vector space to every $B$ homeomorphic to the $n$-ball. |
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These vector spaces glue together associatively, and we require that there is an action of the homeomorphism groupoid. |
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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 |
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homeomorphisms extends to a suitably defined action of the complex of singular chains of homeomorphisms. |
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The axioms for an $A_\infty$ $n$-category are designed to capture two main examples: the blob complexes of $n$-balls labelled by a |
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topological $n$-category, and the complex $\CM{-}{T}$ of maps to a fixed target space $T$. |
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In \S \ref{ss:ncat_fields} we explain how to construct a system of fields from a topological $n$-category |
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(using a colimit along cellulations of a manifold), and in \S \ref{sec:ainfblob} give an alternative definition |
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of the blob complex for an $A_\infty$ $n$-category on an $n$-manifold (analogously, using a homotopy colimit). |
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Using these definitions, we show how to use the blob complex to `resolve' any topological $n$-category as an |
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$A_\infty$ $n$-category, and relate the first and second definitions of the blob complex. |
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We use the blob complex for $A_\infty$ $n$-categories to establish important properties of the blob complex (in both variants), |
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in particular the `gluing formula' of Theorem \ref{thm:gluing} below. |
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The relationship between all these ideas is sketched in Figure \ref{fig:outline}. |
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\tikzstyle{box} = [rectangle, rounded corners, draw,outer sep = 5pt, inner sep = 5pt, line width=0.5pt] |
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\begin{figure}[!ht] |
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{\center |
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\begin{tikzpicture}[align=center,line width = 1.5pt] |
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\newcommand{\xa}{2} |
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\newcommand{\xb}{10} |
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\newcommand{\ya}{14} |
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\newcommand{\yb}{10} |
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\newcommand{\yc}{6} |
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\node[box] at (\xa,\ya) (C) {$\cC$ \\ a topological \\ $n$-category}; |
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\node[box] at (\xb,\ya) (A) {$\underrightarrow{\cC}(M)$ \\ the (dual) TQFT \\ Hilbert space}; |
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\node[box] at (\xa,\yb) (FU) {$(\cF, \cU)$ \\ fields and\\ local relations}; |
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\node[box] at (\xb,\yb) (BC) {$\bc_*(M; \cC)$ \\ the blob complex}; |
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\node[box] at (\xa,\yc) (Cs) {$\cC_*$ \\ an $A_\infty$ \\$n$-category}; |
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\node[box] at (\xb,\yc) (BCs) {$\underrightarrow{\cC_*}(M)$}; |
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\draw[->] (C) -- node[above] {$\displaystyle \colim_{\cell(M)} \cC$} node[below] {\S\S \ref{sec:constructing-a-tqft} \& \ref{ss:ncat_fields}} (A); |
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\draw[->] (FU) -- node[below] {blob complex \\ for $M$} (BC); |
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\draw[->] (Cs) -- node[above] {$\displaystyle \hocolim_{\cell(M)} \cC_*$} node[below] {\S \ref{ss:ncat_fields}} (BCs); |
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\draw[->] (FU) -- node[right=10pt] {$\cF(M)/\cU$} (A); |
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\draw[->] (C) -- node[left=10pt] { |
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Example \ref{ex:traditional-n-categories(fields)} \\ and \S \ref{ss:ncat_fields} |
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%$\displaystyle \cF(M) = \DirectSum_{c \in\cell(M)} \cC(c)$ \\ $\displaystyle \cU(B) = \DirectSum_{c \in \cell(B)} \ker \ev: \cC(c) \to \cC(B)$ |
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} (FU); |
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\draw[->] (BC) -- node[left] {$H_0$} node[right] {c.f. Theorem \ref{thm:skein-modules}} (A); |
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\draw[->] (FU) -- node[left] {blob complex \\ for balls} (Cs); |
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\draw (BC) -- node[right] {$\iso$ by \\ Corollary \ref{cor:new-old}} (BCs); |
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\end{tikzpicture} |
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} |
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\caption{The main gadgets and constructions of the paper.} |
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\label{fig:outline} |
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\end{figure} |
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Finally, later sections address other topics. |
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Section \S \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra, |
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thought of as a topological $n$-category, in terms of the topology of $M$. |
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Section \S \ref{sec:deligne} states (and in a later edition of this paper, hopefully proves) |
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a higher dimensional generalization of the Deligne conjecture (that the little discs operad acts on Hochschild cohomology) in terms of the blob complex. |
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The appendixes prove technical results about $\CH{M}$ and the `small blob complex', |
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and make connections between our definitions of $n$-categories and familiar definitions for $n=1$ and $n=2$, |
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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 |
167 |
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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|
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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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\[ |
185 |
\text{linear combinations of fields} \;\big/\; \text{local relations} , |
|
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\] |
|
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with an appropriately free resolution (the ``blob complex") |
|
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\[ |
|
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\cdots\to \bc_2(W, L) \to \bc_1(W, L) \to \bc_0(W, L) . |
|
190 |
\] |
|
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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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|
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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 |
98 | 198 |
to later papers. |
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||
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\subsection{Formal properties} |
203 |
\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. |
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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 |
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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 $\bc_*^\cC(X)$; |
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this action is extended to all of $C_*(\Homeo(X))$ in Theorem \ref{thm:evaluation} below. |
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The blob complex is also functorial (indeed, exact) with respect to $\cC$, |
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although we will not address this in detail here. |
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98 | 223 |
\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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||
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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, |
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write $X_\text{glued} = X_\text{cut} \bigcup_{Y}\selfarrow$ 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}% |
136 | 236 |
%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 |
|
131 | 241 |
a natural map |
242 |
\[ |
|
187 | 243 |
\bc_*(X_\mathrm{cut}) \to \bc_*(X_\mathrm{glued}) |
131 | 244 |
\] |
187 | 245 |
(natural with respect to homeomorphisms, and also associative with respect to iterated gluings). |
98 | 246 |
\end{property} |
247 |
||
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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 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 associated by the system of fields $\cC$ to balls. |
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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)} |
98 | 254 |
\end{equation} |
255 |
\end{property} |
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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 \ref{property:contractibility} are established in \S \ref{sec:basic-properties}. |
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\subsection{Specializations} |
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\label{sec:specializations} |
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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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265 |
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\begin{thm}[Skein modules] |
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\label{thm:skein-modules}% |
98 | 268 |
The $0$-th blob homology of $X$ is the usual |
269 |
(dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
|
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by $\cC$. |
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(See \S \ref{sec:local-relations}.) |
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\begin{equation*} |
131 | 273 |
H_0(\bc_*^{\cC}(X)) \iso A^{\cC}(X) |
98 | 274 |
\end{equation*} |
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\end{thm} |
98 | 276 |
|
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\newtheorem*{thm:hochschild}{Theorem \ref{thm:hochschild}} |
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\begin{thm:hochschild}[Hochschild homology when $X=S^1$] |
98 | 280 |
The blob complex for a $1$-category $\cC$ on the circle is |
281 |
quasi-isomorphic to the Hochschild complex. |
|
282 |
\begin{equation*} |
|
187 | 283 |
\xymatrix{\bc_*^{\cC}(S^1) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & \HC_*(\cC).} |
98 | 284 |
\end{equation*} |
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\end{thm:hochschild} |
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286 |
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Theorem \ref{thm:skein-modules} is immediate from the definition, and |
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Theorem \ref{thm:hochschild} is established in \S \ref{sec:hochschild}. |
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We also note Appendix \ref{sec:comm_alg} which describes the blob complex when $\cC$ is a one of certain commutative algebras thought of as an $n$-category. |
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\subsection{Structure of the blob complex} |
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\label{sec:structure} |
98 | 294 |
|
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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{thm}[$C_*(\Homeo(-))$ action]\mbox{}\\ |
313 | 297 |
\vspace{-0.5cm} |
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\label{thm:evaluation}% |
313 | 299 |
\begin{enumerate} |
300 |
\item There is a chain map |
|
98 | 301 |
\begin{equation*} |
166 | 302 |
\ev_X: \CH{X} \tensor \bc_*(X) \to \bc_*(X). |
98 | 303 |
\end{equation*} |
304 |
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313 | 305 |
\item Restricted to $C_0(\Homeo(X))$ this is the action of homeomorphisms described in Property \ref{property:functoriality}. |
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|
313 | 307 |
\item For |
147 | 308 |
any codimension $0$-submanifold $Y \sqcup Y^\text{op} \subset \bdy X$ the following diagram |
313 | 309 |
(using the gluing maps described in Property \ref{property:gluing-map}) commutes (up to homotopy). |
147 | 310 |
\begin{equation*} |
311 |
\xymatrix@C+2cm{ |
|
166 | 312 |
\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) \\ |
313 |
\CH{X} \otimes \bc_*(X) |
|
314 |
\ar[r]_{\ev_{X}} \ar[u]^{\gl^{\Homeo}_Y \otimes \gl_Y} & |
|
147 | 315 |
\bc_*(X) \ar[u]_{\gl_Y} |
316 |
} |
|
317 |
\end{equation*} |
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\item Any such chain map satisfying points 2. and 3. above is unique, up to an iterated homotopy. |
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(That is, any pair of homotopies have a homotopy between them, and so on.) |
313 | 320 |
\item This map is associative, in the sense that the following diagram commutes (up to homotopy). |
321 |
\begin{equation*} |
|
322 |
\xymatrix{ |
|
323 |
\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} \\ |
|
324 |
\CH{X} \tensor \bc_*(X) \ar[r]^{\ev_X} & \bc_*(X) |
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325 |
} |
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326 |
\end{equation*} |
|
327 |
\end{enumerate} |
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\end{thm} |
98 | 329 |
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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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In \S \ref{sec:ncats} we introduce the notion of topological $n$-categories, from which we can construct systems of fields. |
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Below, we talk about the blob complex associated to a topological $n$-category, implicitly passing first to the system of fields. |
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Further, in \S \ref{sec:ncats} we also have the notion of an $A_\infty$ $n$-category. |
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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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There is a version of the blob complex for $\cC$ an $A_\infty$ $n$-category |
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instead of a topological $n$-category; this is described in \S \ref{sec:ainfblob}. |
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The definition is in fact simpler, almost tautological, and we use a different notation, $\cl{\cC}(M)$. |
136 | 358 |
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\newtheorem*{thm:product}{Theorem \ref{thm:product}} |
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\begin{thm:product}[Product formula] |
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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 Theorem \ref{thm:blobs-ainfty}). |
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Then |
366 |
\[ |
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\bc_*(Y\times W; \cC) \simeq \cl{\bc_*(Y;\cC)}(W). |
136 | 368 |
\] |
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\end{thm:product} |
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We also give a generalization of this statement for arbitrary fibre bundles, in \S \ref{moddecss}, and a sketch of a statement for arbitrary maps. |
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371 |
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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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(See Appendix \ref{sec:comparing-A-infty} for the translation between topological $A_\infty$ $1$-categories and the usual algebraic notion of an $A_\infty$ category.) |
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\newtheorem*{thm:gluing}{Theorem \ref{thm:gluing}} |
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377 |
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\begin{thm:gluing}[Gluing formula] |
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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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136 | 384 |
\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 $\bc_*(Y)$-bimodule: |
98 | 386 |
\begin{equation*} |
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\bc_*(X_\text{glued}) \simeq \bc_*(X_\text{cut}) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y)}} \selfarrow |
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\end{equation*} |
389 |
\end{itemize} |
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\end{thm:gluing} |
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391 |
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Theorem \ref{thm:evaluation} is proved in |
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in \S \ref{sec:evaluation}, Theorem \ref{thm:blobs-ainfty} appears as Example \ref{ex:blob-complexes-of-balls} in \S \ref{sec:ncats}, |
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and Theorem \ref{thm:product} is proved in \S \ref{ss:product-formula}, with Theorem \ref{thm:gluing} then a relatively straightforward consequence of the proof, explained in \S \ref{sec:gluing}. |
98 | 395 |
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\subsection{Applications} |
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\label{sec:applications} |
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Finally, we give two theorems which we consider as applications. |
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\newtheorem*{thm:map-recon}{Theorem \ref{thm:map-recon}} |
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401 |
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\begin{thm:map-recon}[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{thm:map-recon} |
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This says that we can recover the (homotopic) space of maps to $T$ via blob homology from local data. The proof appears in \S \ref{sec:map-recon}. |
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\newtheorem*{thm:deligne}{Theorem \ref{thm:deligne}} |
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413 |
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\begin{thm:deligne}[Higher dimensional Deligne conjecture] |
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The singular chains of the $n$-dimensional fat graph operad act on blob cochains. |
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\end{thm:deligne} |
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See \S \ref{sec:deligne} for a full explanation of the statement, and an outline of the proof. |
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419 |
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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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We could presumably also replace many of our chain complexes with topological spaces (or indeed, work at the generality of model categories), |
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and likely it will prove useful to think about the connections between what we do here and $(\infty,k)$-categories. |
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More could be said about finite characteristic |
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(there appears in be $2$-torsion in $\bc_1(S^2, \cC)$ for any spherical $2$-category $\cC$, for example). |
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Much more could be said about other types of manifolds, in particular oriented, |
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$\operatorname{Spin}$ and $\operatorname{Pin}^{\pm}$ manifolds, where boundary issues become more complicated. |
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(We'd recommend thinking about boundaries as germs, rather than just codimension $1$ manifolds.) |
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We've also take the path of least resistance by considering $\operatorname{PL}$ manifolds; |
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there may be some differences for topological manifolds and smooth manifolds. |
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The paper ``Skein homology'' \cite{MR1624157} has similar motivations, and it may be |
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interesting to investigate if there is a connection with the material here. |
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Many results in Hochschild homology can be understood `topologically' via the blob complex. |
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For example, we expect that the shuffle product on the Hochschild homology of a commutative algebra $A$ |
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(see \cite[\S 4.2]{MR1600246}) simply corresponds to the gluing operation on $\bc_*(S^1 \times [0,1], A)$, |
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but haven't investigated the details. |
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Most importantly, however, \nn{applications!} \nn{cyclic homology, $n=2$ cases, contact, Kh} |
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\subsection{Thanks and acknowledgements} |
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We'd like to thank David Ben-Zvi, Kevin Costello, Chris Douglas, |
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Michael Freedman, Vaughan Jones, Alexander Kirillov, Justin Roberts, Chris Schommer-Pries, Peter Teichner, Thomas Tradler \nn{and who else?} for many interesting and useful conversations. |
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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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