576 that is, a permissible decomposition $x$ of $W$ together with an element of $\psi_{\cC;W}(x)$. |
576 that is, a permissible decomposition $x$ of $W$ together with an element of $\psi_{\cC;W}(x)$. |
577 |
577 |
578 |
578 |
579 \subsubsection{Colimits} |
579 \subsubsection{Colimits} |
580 Recall that our definition of an $n$-category is essentially a collection of functors |
580 Recall that our definition of an $n$-category is essentially a collection of functors |
581 defined on the categories of homeomorphisms $k$-balls |
581 defined on the categories of homeomorphisms of $k$-balls |
582 for $k \leq n$ satisfying certain axioms. |
582 for $k \leq n$ satisfying certain axioms. |
583 It is natural to hope to extend such functors to the |
583 It is natural to hope to extend such functors to the |
584 larger categories of all $k$-manifolds (again, with homeomorphisms). |
584 larger categories of all $k$-manifolds (again, with homeomorphisms). |
585 In fact, the axioms stated above already require such an extension to $k$-spheres for $k<n$. |
585 In fact, the axioms stated above already require such an extension to $k$-spheres for $k<n$. |
586 |
586 |
587 The natural construction achieving this is a colimit along the poset of permissible decompositions. |
587 The natural construction achieving this is a colimit along the poset of permissible decompositions. |
588 For an isotopy $n$-category $\cC$, |
588 Given an isotopy $n$-category $\cC$, |
589 we will denote the extension to all manifolds by $\cl{\cC}$. On a $k$-manifold $W$, with $k \leq n$, |
589 we will denote its extension to all manifolds by $\cl{\cC}$. On a $k$-manifold $W$, with $k \leq n$, |
590 this is defined to be the colimit along $\cell(W)$ of the functor $\psi_{\cC;W}$. |
590 this is defined to be the colimit along $\cell(W)$ of the functor $\psi_{\cC;W}$. |
591 Note that Axioms \ref{axiom:composition} and \ref{axiom:associativity} |
591 Note that Axioms \ref{axiom:composition} and \ref{axiom:associativity} |
592 imply that $\cl{\cC}(X) \iso \cC(X)$ when $X$ is a $k$-ball with $k<n$. |
592 imply that $\cl{\cC}(X) \iso \cC(X)$ when $X$ is a $k$-ball with $k<n$. |
593 Recall that given boundary conditions $c \in \cl{\cC}(\bdy X)$, for $X$ an $n$-ball, |
593 Recall that given boundary conditions $c \in \cl{\cC}(\bdy X)$, for $X$ an $n$-ball, |
594 the set $\cC(X;c)$ is a vector space. Using this, we note that for $c \in \cl{\cC}(\bdy W)$, |
594 the set $\cC(X;c)$ is a vector space (we assume $\cC$ is enriched in vector spaces). |
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595 Using this, we note that for $c \in \cl{\cC}(\bdy W)$, |
595 for $W$ an arbitrary $n$-manifold, the set $\cl{\cC}(W;c) = \bdy^{-1} (c)$ inherits the structure of a vector space. |
596 for $W$ an arbitrary $n$-manifold, the set $\cl{\cC}(W;c) = \bdy^{-1} (c)$ inherits the structure of a vector space. |
596 These are the usual TQFT skein module invariants on $n$-manifolds. |
597 These are the usual TQFT skein module invariants on $n$-manifolds. |
597 |
598 |
598 We can now give a straightforward but rather abstract definition of the blob complex of an $n$-manifold $W$ |
599 We can now give a straightforward but rather abstract definition of the blob complex of an $n$-manifold $W$ |
599 with coefficients in the $n$-category $\cC$ as the homotopy colimit along $\cell(W)$ |
600 with coefficients in the $n$-category $\cC$ as the {\it homotopy} colimit along $\cell(W)$ |
600 of the functor $\psi_{\cC; W}$ described above. We write this as $\clh{\cC}(W)$. |
601 of the functor $\psi_{\cC; W}$ described above. We write this as $\clh{\cC}(W)$. |
601 |
602 |
602 An explicit realization of the homotopy colimit is provided by the simplices of the |
603 An explicit realization of the homotopy colimit is provided by the simplices of the |
603 functor $\psi_{\cC; W}$. That is, $$\clh{\cC}(W) = \DirectSum_{\bar{x}} \psi_{\cC; W}(x_0)[m],$$ |
604 functor $\psi_{\cC; W}$. That is, $$\clh{\cC}(W) = \DirectSum_{\bar{x}} \psi_{\cC; W}(x_0)[m],$$ |
604 where $\bar{x} = x_0 \leq \cdots \leq x_m$ is a simplex in $\cell(W)$. |
605 where $\bar{x} = x_0 \leq \cdots \leq x_m$ is a simplex in $\cell(W)$. |
611 A cone-product polyhedra is obtained from a point by successively taking the cone or taking the |
612 A cone-product polyhedra is obtained from a point by successively taking the cone or taking the |
612 product with another cone-product polyhedron. Just as simplices correspond to linear directed graphs, |
613 product with another cone-product polyhedron. Just as simplices correspond to linear directed graphs, |
613 cone-product polyheda correspond to directed trees: taking cone adds a new root before the existing root, |
614 cone-product polyheda correspond to directed trees: taking cone adds a new root before the existing root, |
614 and taking product identifies the roots of several trees. |
615 and taking product identifies the roots of several trees. |
615 The ``local homotopy colimit" is then defined according to the same formula as above, but with $\bar{x}$ a cone-product polyhedron in $\cell(W)$. |
616 The ``local homotopy colimit" is then defined according to the same formula as above, but with $\bar{x}$ a cone-product polyhedron in $\cell(W)$. |
|
617 We further require that any morphism in a directed tree is not expressible as a product. |
616 The differential acts on $(\bar{x},a)$ both on $a$ and on $\bar{x}$, applying the appropriate gluing map to $a$ when required. |
618 The differential acts on $(\bar{x},a)$ both on $a$ and on $\bar{x}$, applying the appropriate gluing map to $a$ when required. |
617 A Eilenberg-Zilber subdivision argument shows this is the same as the usual realization. |
619 A Eilenberg-Zilber subdivision argument shows this is the same as the usual realization. |
618 In the local homotopy colimit we can further require that any composition of morphisms in a directed tree is not expressible as a product. |
|
619 |
620 |
620 %When $\cC$ is a topological $n$-category, |
621 %When $\cC$ is a topological $n$-category, |
621 %the flexibility available in the construction of a homotopy colimit allows |
622 %the flexibility available in the construction of a homotopy colimit allows |
622 %us to give a much more explicit description of the blob complex which we'll write as $\bc_*(W; \cC)$. |
623 %us to give a much more explicit description of the blob complex which we'll write as $\bc_*(W; \cC)$. |
623 %\todo{either need to explain why this is the same, or significantly rewrite this section} |
624 %\todo{either need to explain why this is the same, or significantly rewrite this section} |
632 The next few paragraphs describe this in more detail. |
633 The next few paragraphs describe this in more detail. |
633 |
634 |
634 We say a collection of balls $\{B_i\}$ in a manifold $W$ is \emph{permissible} |
635 We say a collection of balls $\{B_i\}$ in a manifold $W$ is \emph{permissible} |
635 if there exists a permissible decomposition $M_0\to\cdots\to M_m = W$ such that |
636 if there exists a permissible decomposition $M_0\to\cdots\to M_m = W$ such that |
636 each $B_i$ appears as a connected component of one of the $M_j$. |
637 each $B_i$ appears as a connected component of one of the $M_j$. |
637 Note that this allows the balls to be pairwise either disjoint or nested. |
638 Note that this forces the balls to be pairwise either disjoint or nested. |
638 Such a collection of balls cuts $W$ into pieces, the connected components of $W \setminus \bigcup \bdy B_i$. |
639 Such a collection of balls cuts $W$ into pieces, the connected components of $W \setminus \bigcup \bdy B_i$. |
639 These pieces need not be manifolds, but they do automatically have permissible decompositions. |
640 These pieces need not be manifolds, but they do automatically have permissible decompositions. |
640 |
641 |
641 The $k$-blob group $\bc_k(W; \cC)$ is generated by the $k$-blob diagrams. A $k$-blob diagram consists of |
642 The $k$-blob group $\bc_k(W; \cC)$ is generated by the $k$-blob diagrams. A $k$-blob diagram consists of |
642 \begin{itemize} |
643 \begin{itemize} |
727 |
728 |
728 The blob complex has several important special cases. |
729 The blob complex has several important special cases. |
729 |
730 |
730 \begin{thm}[Skein modules] |
731 \begin{thm}[Skein modules] |
731 \label{thm:skein-modules} |
732 \label{thm:skein-modules} |
732 Suppose $\cC$ is an isotopy $n$-category |
733 Suppose $\cC$ is an isotopy $n$-category. |
733 The $0$-th blob homology of $X$ is the usual |
734 The $0$-th blob homology of $X$ is the usual |
734 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
735 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
735 by $\cC$. |
736 by $\cC$. |
736 \begin{equation*} |
737 \begin{equation*} |
737 H_0(\bc_*(X;\cC)) \iso \cl{\cC}(X) |
738 H_0(\bc_*(X;\cC)) \iso \cl{\cC}(X) |
761 \end{thm} |
762 \end{thm} |
762 |
763 |
763 This says that we can recover (up to homotopy) the space of maps to $T$ via blob homology from local data. |
764 This says that we can recover (up to homotopy) the space of maps to $T$ via blob homology from local data. |
764 Note that there is no restriction on the connectivity of $T$ as there is for |
765 Note that there is no restriction on the connectivity of $T$ as there is for |
765 the corresponding result in topological chiral homology \cite[Theorem 3.8.6]{0911.0018}. |
766 the corresponding result in topological chiral homology \cite[Theorem 3.8.6]{0911.0018}. |
766 The result was proved in \cite[\S 7.3]{1009.5025}. |
767 The result is proved in \cite[\S 7.3]{1009.5025}. |
767 |
768 |
768 \subsection{Structure of the blob complex} |
769 \subsection{Structure of the blob complex} |
769 \label{sec:structure} |
770 \label{sec:structure} |
770 |
771 |
771 In the following $\CH{X} = C_*(\Homeo(X))$ is the singular chain complex of the space |
772 In the following $\CH{X} = C_*(\Homeo(X))$ is the singular chain complex of the space |
836 which can be thought of as a free resolution. |
837 which can be thought of as a free resolution. |
837 \end{rem} |
838 \end{rem} |
838 This result is described in more detail as Example 6.2.8 of \cite{1009.5025}. |
839 This result is described in more detail as Example 6.2.8 of \cite{1009.5025}. |
839 |
840 |
840 Fix a topological $n$-category $\cC$, which we'll now omit from notation. |
841 Fix a topological $n$-category $\cC$, which we'll now omit from notation. |
841 Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ category. |
842 Recall (Theorem \ref{thm:blobs-ainfty}) that there is associated to |
|
843 any $(n{-}1)$-manifold $Y$ an $A_\infty$ category $\bc_*(Y)$. |
842 |
844 |
843 \begin{thm}[Gluing formula] |
845 \begin{thm}[Gluing formula] |
844 \label{thm:gluing} |
846 \label{thm:gluing} |
845 \mbox{}\vspace{-0.2cm}% <-- gets the indenting right |
847 \mbox{}\vspace{-0.2cm}% <-- gets the indenting right |
846 \begin{itemize} |
848 \begin{itemize} |
847 \item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, |
849 \item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, |
848 the blob complex of $X$ is naturally an |
850 the blob complex of $X$ is naturally an |
849 $A_\infty$ module for $\bc_*(Y)$. |
851 $A_\infty$ module for $\bc_*(Y)$. |
850 |
852 |
851 \item The blob complex of a glued manifold $X\bigcup_Y \selfarrow$ is the $A_\infty$ self-tensor product of |
853 \item The blob complex of a glued manifold $X\bigcup_Y \selfarrow$ is the $A_\infty$ self-tensor product of |
852 $\bc_*(X)$ as an $\bc_*(Y)$-bimodule: |
854 $\bc_*(X)$ as a $\bc_*(Y)$-bimodule: |
853 \begin{equation*} |
855 \begin{equation*} |
854 \bc_*(X\bigcup_Y \selfarrow) \simeq \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y)}} \selfarrow |
856 \bc_*(X\bigcup_Y \selfarrow) \simeq \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y)}} \selfarrow |
855 \end{equation*} |
857 \end{equation*} |
856 \end{itemize} |
858 \end{itemize} |
857 \end{thm} |
859 \end{thm} |
862 |
864 |
863 Let $T_*$ denote the self tensor product of $\bc_*(X)$, which is a homotopy colimit. |
865 Let $T_*$ denote the self tensor product of $\bc_*(X)$, which is a homotopy colimit. |
864 There is a tautological map from the 0-simplices of $T_*$ to $\bc_*(X\bigcup_Y \selfarrow)$, |
866 There is a tautological map from the 0-simplices of $T_*$ to $\bc_*(X\bigcup_Y \selfarrow)$, |
865 and this map can be extended to a chain map on all of $T_*$ by sending the higher simplices to zero. |
867 and this map can be extended to a chain map on all of $T_*$ by sending the higher simplices to zero. |
866 Constructing a homotopy inverse to this natural map involves making various choices, but one can show that the |
868 Constructing a homotopy inverse to this natural map involves making various choices, but one can show that the |
867 choices form contractible subcomplexes and apply the theory of acyclic models. |
869 choices form contractible subcomplexes and apply the acyclic models theorem. |
868 \end{proof} |
870 \end{proof} |
869 |
871 |
870 We next describe the blob complex for product manifolds, in terms of the |
872 We next describe the blob complex for product manifolds, in terms of the |
871 blob complexes for the $A_\infty$ $n$-categories constructed as above. |
873 blob complexes for the $A_\infty$ $n$-categories constructed as above. |
872 |
874 |
877 Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ as above. |
879 Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ as above. |
878 Then |
880 Then |
879 \[ |
881 \[ |
880 \bc_*(Y\times W; \cC) \simeq \clh{\bc_*(Y;\cC)}(W). |
882 \bc_*(Y\times W; \cC) \simeq \clh{\bc_*(Y;\cC)}(W). |
881 \] |
883 \] |
|
884 That is, the blob complex of $Y\times W$ with coefficients in $\cC$ is homotopy equivalent |
|
885 to the blob complex of $W$ with coefficients in $\bc_*(Y;\cC)$. |
882 \end{thm} |
886 \end{thm} |
883 The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps |
887 The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps |
884 (see \cite[\S7.1]{1009.5025}). |
888 (see \cite[\S7.1]{1009.5025}). |
885 |
889 |
886 \begin{proof} (Sketch.) |
890 \begin{proof} (Sketch.) |
923 Recall (Theorem \ref{thm:evaluation}) that chains on the space of mapping cylinders also act on the |
927 Recall (Theorem \ref{thm:evaluation}) that chains on the space of mapping cylinders also act on the |
924 blob complex. |
928 blob complex. |
925 An $n$-dimensional surgery cylinder is |
929 An $n$-dimensional surgery cylinder is |
926 defined to be a sequence of mapping cylinders and surgeries (Figure \ref{delfig2}), |
930 defined to be a sequence of mapping cylinders and surgeries (Figure \ref{delfig2}), |
927 modulo changing the order of distant surgeries, and conjugating a submanifold not modified in a surgery by a homeomorphism. |
931 modulo changing the order of distant surgeries, and conjugating a submanifold not modified in a surgery by a homeomorphism. |
928 One can associated to this data an $(n{+}1)$-manifold with a foliation by intervals, |
932 One can associate to this data an $(n{+}1)$-manifold with a foliation by intervals, |
929 and the relations we impose correspond to homeomorphisms of the $(n{+}1)$-manifolds |
933 and the relations we impose correspond to homeomorphisms of the $(n{+}1)$-manifolds |
930 which preserve the foliation. |
934 which preserve the foliation. |
931 |
935 |
932 Surgery cylinders form an operad, by gluing the outer boundary of one cylinder into an inner boundary of another. |
936 Surgery cylinders form an operad, by gluing the outer boundary of one cylinder into an inner boundary of another. |
933 |
937 |