284 homeomorphisms which are not the identity on the boundary of the $k$-ball. |
284 homeomorphisms which are not the identity on the boundary of the $k$-ball. |
285 The action of these homeomorphisms gives the ``strong duality" structure. |
285 The action of these homeomorphisms gives the ``strong duality" structure. |
286 As such, we don't subdivide the boundary of a morphism |
286 As such, we don't subdivide the boundary of a morphism |
287 into domain and range --- the duality operations can convert between domain and range. |
287 into domain and range --- the duality operations can convert between domain and range. |
288 |
288 |
289 Later \todo{} we inductively define an extension of the functors $\cC_k$ to functors $\cl{\cC}_k$ from arbitrary manifolds to sets. We need the restriction of these functors to $k$-spheres, for $k<n$, for the next axiom. |
289 Later \nn{make sure this actually happens, or reorganise} we inductively define an extension of the functors $\cC_k$ to functors $\cl{\cC}_k$ from arbitrary manifolds to sets. We need the restriction of these functors to $k$-spheres, for $k<n$, for the next axiom. |
290 |
290 |
291 \begin{axiom}[Boundaries]\label{nca-boundary} |
291 \begin{axiom}[Boundaries]\label{nca-boundary} |
292 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$. |
292 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$. |
293 These maps, for various $X$, comprise a natural transformation of functors. |
293 These maps, for various $X$, comprise a natural transformation of functors. |
294 \end{axiom} |
294 \end{axiom} |
526 that is, a permissible decomposition $x$ of $W$ together with an element of $\psi_{\cC;W}(x)$. |
526 that is, a permissible decomposition $x$ of $W$ together with an element of $\psi_{\cC;W}(x)$. |
527 |
527 |
528 |
528 |
529 \subsubsection{Homotopy colimits} |
529 \subsubsection{Homotopy colimits} |
530 \nn{Motivation: How can we extend an $n$-category from balls to arbitrary manifolds?} |
530 \nn{Motivation: How can we extend an $n$-category from balls to arbitrary manifolds?} |
531 \todo{Mention that the axioms for $n$-categories can be stated in terms of decompositions of balls?} |
531 \nn{Mention that the axioms for $n$-categories can be stated in terms of decompositions of balls?} |
532 \nn{Explain codimension colimits here too} |
532 \nn{Explain codimension colimits here too} |
533 |
533 |
534 We can now give a straightforward but rather abstract definition of the blob complex of an $n$-manifold $W$ |
534 We can now give a straightforward but rather abstract definition of the blob complex of an $n$-manifold $W$ |
535 with coefficients in the $n$-category $\cC$ as the homotopy colimit along $\cell(W)$ |
535 with coefficients in the $n$-category $\cC$ as the homotopy colimit along $\cell(W)$ |
536 of the functor $\psi_{\cC; W}$ described above. We write this as $\clh{\cC}(W)$. |
536 of the functor $\psi_{\cC; W}$ described above. We write this as $\clh{\cC}(W)$. |
537 |
537 |
538 An explicit realization of the homotopy colimit is provided by the simplices of the functor $\psi_{\cC; W}$. That is, $$\clh{\cC}(W) = \DirectSum_{\bar{x}} \psi_{\cC; W}(x_0)[m],$$ where $\bar{x} = x_0 \leq \cdots \leq x_m$ is a simplex in $\cell(W)$. The differential acts on $(\bar{x},a)$ (here $a \in \psi_{\cC; W}(x_0)$) as |
538 An explicit realization of the homotopy colimit is provided by the simplices of the functor $\psi_{\cC; W}$. That is, $$\clh{\cC}(W) = \DirectSum_{\bar{x}} \psi_{\cC; W}(x_0)[m],$$ where $\bar{x} = x_0 \leq \cdots \leq x_m$ is a simplex in $\cell(W)$. The differential acts on $(\bar{x},a)$ (here $a \in \psi_{\cC; W}(x_0)$) as |
539 $$\bdy (\bar{x},a) = (\bar{x}, \bdy a) + (-1)^{\deg a} \left( (d_0 \bar{x}, g(a)) + \sum_{i=1}^m (-1)^i (d_i \bar{x}, a) \right)$$ |
539 $$\bdy (\bar{x},a) = (\bar{x}, \bdy a) + (-1)^{\deg a} \left( (d_0 \bar{x}, g(a)) + \sum_{i=1}^m (-1)^i (d_i \bar{x}, a) \right)$$ |
540 where $g$ is the gluing map from $x_0$ to $x_1$, and $d_i \bar{x}$ denotes the $i$-th face of the simplex $\bar{x}$. |
540 where $g$ is the gluing map from $x_0$ to $x_1$, and $d_i \bar{x}$ denotes the $i$-th face of the simplex $\bar{x}$. |
541 |
541 |
542 Alternatively, we can take advantage of the product structure on $\cell(W)$ to realize the homotopy colimit via the cone-product polyhedra in $\cell(W)$. A cone-product polyhedra is obtained from a point by successively taking the cone or taking the product with another cone-product polyhedron. Just as simplices correspond to linear directed graphs, cone-product polyheda correspond to directed trees: taking cone adds a new root before the existing root, and taking product identifies the roots of several trees. The `local homotopy colimit' is then defined according to the same formula as above, but with $x$ a cone-product polyhedron in $\cell(W)$. |
542 Alternatively, we can take advantage of the product structure on $\cell(W)$ to realize the homotopy colimit via the cone-product polyhedra in $\cell(W)$. A cone-product polyhedra is obtained from a point by successively taking the cone or taking the product with another cone-product polyhedron. Just as simplices correspond to linear directed graphs, cone-product polyheda correspond to directed trees: taking cone adds a new root before the existing root, and taking product identifies the roots of several trees. The `local homotopy colimit' is then defined according to the same formula as above, but with $\bar{x}$ a cone-product polyhedron in $\cell(W)$. The differential acts on $(\bar{x},a)$ both on $a$ and on $\bar{x}$, applying the appropriate gluing map to $a$ when required. |
543 A Eilenberg-Zilber subdivision argument shows this is the same as the usual realization. |
543 A Eilenberg-Zilber subdivision argument shows this is the same as the usual realization. |
544 |
544 |
545 %When $\cC$ is a topological $n$-category, |
545 %When $\cC$ is a topological $n$-category, |
546 %the flexibility available in the construction of a homotopy colimit allows |
546 %the flexibility available in the construction of a homotopy colimit allows |
547 %us to give a much more explicit description of the blob complex which we'll write as $\bc_*(W; \cC)$. |
547 %us to give a much more explicit description of the blob complex which we'll write as $\bc_*(W; \cC)$. |
659 quasi-isomorphic to the Hochschild complex. |
659 quasi-isomorphic to the Hochschild complex. |
660 \begin{equation*} |
660 \begin{equation*} |
661 \xymatrix{\bc_*(S^1;\cC) \ar[r]^(0.47){\iso}_(0.47){\text{qi}} & \HC_*(\cC).} |
661 \xymatrix{\bc_*(S^1;\cC) \ar[r]^(0.47){\iso}_(0.47){\text{qi}} & \HC_*(\cC).} |
662 \end{equation*} |
662 \end{equation*} |
663 \end{thm} |
663 \end{thm} |
664 |
664 This theorem is established by extending the statement to bimodules as well as categories, then verifying that the universal properties of Hochschild homology also hold for $\bc_*(S^1; -)$. |
665 Theorem \ref{thm:skein-modules} is immediate from the definition, and |
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666 Theorem \ref{thm:hochschild} is established by extending the statement to bimodules as well as categories, then verifying that the universal properties of Hochschild homology also hold for $\bc_*(S^1; -)$. |
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667 |
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668 |
665 |
669 \begin{thm}[Mapping spaces] |
666 \begin{thm}[Mapping spaces] |
670 \label{thm:map-recon} |
667 \label{thm:map-recon} |
671 Let $\pi^\infty_{\le n}(T)$ denote the $A_\infty$ $n$-category based on maps |
668 Let $\pi^\infty_{\le n}(T)$ denote the $A_\infty$ $n$-category based on maps |
672 $B^n \to T$. |
669 $B^n \to T$. |
674 Then |
671 Then |
675 $$\bc_*(X; \pi^\infty_{\le n}(T)) \simeq \CM{X}{T}.$$ |
672 $$\bc_*(X; \pi^\infty_{\le n}(T)) \simeq \CM{X}{T}.$$ |
676 \end{thm} |
673 \end{thm} |
677 |
674 |
678 This says that we can recover (up to homotopy) the space of maps to $T$ via blob homology from local data. |
675 This says that we can recover (up to homotopy) the space of maps to $T$ via blob homology from local data. |
679 Note that there is no restriction on the connectivity of $T$ as there is for the corresponding result in topological chiral homology \cite[Theorem 3.8.6]{0911.0018}. |
676 Note that there is no restriction on the connectivity of $T$ as there is for the corresponding result in topological chiral homology \cite[Theorem 3.8.6]{0911.0018}. The result was proved in \cite[\S 7.3]{1009.5025}. |
680 \todo{sketch proof} |
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681 |
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682 |
677 |
683 \subsection{Structure of the blob complex} |
678 \subsection{Structure of the blob complex} |
684 \label{sec:structure} |
679 \label{sec:structure} |
685 |
680 |
686 In the following $\CH{X} = C_*(\Homeo(X))$ is the singular chain complex of the space of homeomorphisms of $X$, fixed on $\bdy X$. |
681 In the following $\CH{X} = C_*(\Homeo(X))$ is the singular chain complex of the space of homeomorphisms of $X$, fixed on $\bdy X$. |
865 |
860 |
866 Consider the special case where $n=1$ and all of the $M_i$'s and $N_i$'s are 1-balls. |
861 Consider the special case where $n=1$ and all of the $M_i$'s and $N_i$'s are 1-balls. |
867 We have that $SC^1_{\overline{M}, \overline{N}}$ is homotopy equivalent to the little |
862 We have that $SC^1_{\overline{M}, \overline{N}}$ is homotopy equivalent to the little |
868 disks operad and $\hom(\bc_*(M_i), \bc_*(N_i))$ is homotopy equivalent to Hochschild cochains. |
863 disks operad and $\hom(\bc_*(M_i), \bc_*(N_i))$ is homotopy equivalent to Hochschild cochains. |
869 This special case is just the usual Deligne conjecture |
864 This special case is just the usual Deligne conjecture |
870 (see \cite{hep-th/9403055, MR1805894, MR2064592, MR1805923} |
865 (see \cite{hep-th/9403055, MR1328534, MR1805894, MR1805923, MR2064592}). |
871 \nn{should check that this is the optimal list of references; what about Gerstenhaber-Voronov?; |
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872 if we revise this list, should propagate change back to main paper} |
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873 ). |
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874 |
866 |
875 The general case when $n=1$ goes beyond the original Deligne conjecture, as the $M_i$'s and $N_i$'s |
867 The general case when $n=1$ goes beyond the original Deligne conjecture, as the $M_i$'s and $N_i$'s |
876 could be disjoint unions of 1-balls and circles, and the surgery cylinders could be high genus surfaces. |
868 could be disjoint unions of 1-balls and circles, and the surgery cylinders could be high genus surfaces. |
877 |
869 |
878 If all of the $M_i$'s and $N_i$'s are $n$-balls, then $SC^n_{\overline{M}, \overline{N}}$ |
870 If all of the $M_i$'s and $N_i$'s are $n$-balls, then $SC^n_{\overline{M}, \overline{N}}$ |