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%!TEX root = ../blob1.tex
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\section{Introduction}
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[Outline for intro]
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\begin{itemize}
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\item Starting point: TQFTs via fields and local relations.
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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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\item For non-semiemple 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 $4{+}1$-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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\item 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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\nn{... $L_1, L_2, L_3$}.
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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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\item 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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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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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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\item 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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\item The solution is to replace $A_{Kh}(W^4, L)$, which is a quotient
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\[
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\text{linear combinations of fields} \;\big/\; \text{local relations} ,
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\]
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with an appropriately free resolution (the ``blob complex")
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\[
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\cdots\to \bc_2(W, L) \to \bc_1(W, L) \to \bc_0(W, L) .
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\]
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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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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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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$\bc_2$ is linear combinations of relations amongst relations on $W$,
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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and so on.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\item None of the above ideas depend on the details of the Khovanov homology example,
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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so we develop the general theory in the paper and postpone specific applications
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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to later papers.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\item The blob complex enjoys the following nice properties \nn{...}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\end{itemize}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\bigskip
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\hrule
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\bigskip
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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We then show that blob homology enjoys the following
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\ref{property:gluing} properties.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\begin{property}[Functoriality]
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\label{property:functoriality}%
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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Blob homology is functorial with respect to diffeomorphisms. That is, fixing an $n$-dimensional system of fields $\cF$ and local relations $\cU$, the association
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\begin{equation*}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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X \mapsto \bc_*^{\cF,\cU}(X)
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\end{equation*}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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is a functor from $n$-manifolds and diffeomorphisms between them to chain complexes and isomorphisms between them.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\end{property}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\begin{property}[Disjoint union]
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\label{property:disjoint-union}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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The blob complex of a disjoint union is naturally the tensor product of the blob complexes.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\begin{equation*}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\bc_*(X_1 \du X_2) \iso \bc_*(X_1) \tensor \bc_*(X_2)
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\end{equation*}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\end{property}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\begin{property}[A map for gluing]
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\label{property:gluing-map}%
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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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 $\bdy X_2$,
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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there is a chain map
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\begin{equation*}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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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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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\end{equation*}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\end{property}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\begin{property}[Contractibility]
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\label{property:contractibility}%
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\todo{Err, requires a splitting?}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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The blob complex for an $n$-category on an $n$-ball is quasi-isomorphic to its $0$-th homology.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\begin{equation}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\xymatrix{\bc_*^{\cC}(B^n) \ar[r]^{\iso}_{\text{qi}} & H_0(\bc_*^{\cC}(B^n))}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\end{equation}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\todo{Say that this is just the original $n$-category?}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\end{property}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\begin{property}[Skein modules]
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\label{property:skein-modules}%
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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The $0$-th blob homology of $X$ is the usual
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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(dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$
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by $(\cF,\cU)$. (See \S \ref{sec:local-relations}.)
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\begin{equation*}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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H_0(\bc_*^{\cF,\cU}(X)) \iso A^{\cF,\cU}(X)
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\end{equation*}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\end{property}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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112 |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\begin{property}[Hochschild homology when $X=S^1$]
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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114 |
\label{property:hochschild}%
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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The blob complex for a $1$-category $\cC$ on the circle is
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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116 |
quasi-isomorphic to the Hochschild complex.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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117 |
\begin{equation*}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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118 |
\xymatrix{\bc_*^{\cC}(S^1) \ar[r]^{\iso}_{\text{qi}} & HC_*(\cC)}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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119 |
\end{equation*}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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120 |
\end{property}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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121 |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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122 |
\begin{property}[Evaluation map]
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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123 |
\label{property:evaluation}%
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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There is an `evaluation' chain map
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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125 |
\begin{equation*}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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126 |
\ev_X: \CD{X} \tensor \bc_*(X) \to \bc_*(X).
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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127 |
\end{equation*}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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128 |
(Here $\CD{X}$ is the singular chain complex of the space of diffeomorphisms of $X$, fixed on $\bdy X$.)
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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129 |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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Restricted to $C_0(\Diff(X))$ this is just the action of diffeomorphisms described in Property \ref{property:functoriality}. Further, for
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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131 |
any codimension $1$-submanifold $Y \subset X$ dividing $X$ into $X_1 \cup_Y X_2$, the following diagram
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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132 |
(using the gluing maps described in Property \ref{property:gluing-map}) commutes.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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133 |
\begin{equation*}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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134 |
\xymatrix{
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\CD{X} \otimes \bc_*(X) \ar[r]^{\ev_X} & \bc_*(X) \\
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136 |
\CD{X_1} \otimes \CD{X_2} \otimes \bc_*(X_1) \otimes \bc_*(X_2)
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\ar@/_4ex/[r]_{\ev_{X_1} \otimes \ev_{X_2}} \ar[u]^{\gl^{\Diff}_Y \otimes \gl_Y} &
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138 |
\bc_*(X_1) \otimes \bc_*(X_2) \ar[u]_{\gl_Y}
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}
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140 |
\end{equation*}
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\nn{should probably say something about associativity here (or not?)}
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142 |
\end{property}
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143 |
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144 |
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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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148 |
\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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150 |
naturally an $A_\infty$ category. % We'll write $\bc_*(Y)$ for $\bc_*(Y \times I)$ below.
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151 |
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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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153 |
$A_\infty$ module for $\bc_*(Y \times I)$.
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154 |
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\item For any $n$-manifold $X$, with $Y \cup Y^{\text{op}}$ a codimension
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$0$-submanifold of its boundary, the blob homology of $X'$, obtained from
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$X$ by gluing along $Y$, is the $A_\infty$ self-tensor product of
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158 |
$\bc_*(X)$ as an $\bc_*(Y \times I)$-bimodule.
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159 |
\begin{equation*}
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\bc_*(X') \iso \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y \times I)}} \!\!\!\!\!\!\xymatrix{ \ar@(ru,rd)@<-1ex>[]}
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161 |
\end{equation*}
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162 |
\end{itemize}
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163 |
\end{property}
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164 |
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165 |
\nn{add product formula? $n$-dimensional fat graph operad stuff?}
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166 |
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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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168 |
\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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171 |
and Property \ref{property:gluing} in \S \ref{sec:gluing}.
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