622 that it is homotopic to its $0$-th homology, and this is just the vector space associated to the ball by the $n$-category. |
622 that it is homotopic to its $0$-th homology, and this is just the vector space associated to the ball by the $n$-category. |
623 \begin{equation*} |
623 \begin{equation*} |
624 \xymatrix{\bc_*(B^n;\cC) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*(B^n;\cC)) \ar[r]^(0.6)\iso & \cC(B^n)} |
624 \xymatrix{\bc_*(B^n;\cC) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*(B^n;\cC)) \ar[r]^(0.6)\iso & \cC(B^n)} |
625 \end{equation*} |
625 \end{equation*} |
626 \end{property} |
626 \end{property} |
627 \nn{maybe should say something about the $A_\infty$ case} |
627 %\nn{maybe should say something about the $A_\infty$ case} |
628 |
628 |
629 \begin{proof}(Sketch) |
629 \begin{proof}(Sketch) |
630 For $k\ge 1$, the contracting homotopy sends a $k$-blob diagram to the $(k{+}1)$-blob diagram |
630 For $k\ge 1$, the contracting homotopy sends a $k$-blob diagram to the $(k{+}1)$-blob diagram |
631 obtained by adding an outer $(k{+}1)$-st blob consisting of all $B^n$. |
631 obtained by adding an outer $(k{+}1)$-st blob consisting of all $B^n$. |
632 For $k=0$ we choose a splitting $s: H_0(\bc_*(B^n)) \to \bc_0(B^n)$ and send |
632 For $k=0$ we choose a splitting $s: H_0(\bc_*(B^n)) \to \bc_0(B^n)$ and send |
633 $x\in \bc_0(B^n)$ to $x - s([x])$, where $[x]$ denotes the image of $x$ in $H_0(\bc_*(B^n))$. |
633 $x\in \bc_0(B^n)$ to $x - s([x])$, where $[x]$ denotes the image of $x$ in $H_0(\bc_*(B^n))$. |
634 \end{proof} |
634 \end{proof} |
|
635 |
|
636 If $\cC$ is an $A-\infty$ $n$-category then $\bc_*(B^n;\cC)$ is still homotopy equivalent to $\cC(B^n)$, |
|
637 but this is no longer concentrated in degree zero. |
635 |
638 |
636 \subsection{Specializations} |
639 \subsection{Specializations} |
637 \label{sec:specializations} |
640 \label{sec:specializations} |
638 |
641 |
639 The blob complex has several important special cases. |
642 The blob complex has several important special cases. |
823 \] |
826 \] |
824 We think of this map as being associated to a surgery which cuts $M$ out of $M\cup_E R$ and |
827 We think of this map as being associated to a surgery which cuts $M$ out of $M\cup_E R$ and |
825 replaces it with $N$, yielding $N\cup_E R$. |
828 replaces it with $N$, yielding $N\cup_E R$. |
826 (This is a more general notion of surgery that usual --- $M$ and $N$ can be any manifolds |
829 (This is a more general notion of surgery that usual --- $M$ and $N$ can be any manifolds |
827 which share a common boundary.) |
830 which share a common boundary.) |
|
831 In analogy to Hochschild cochains, we will call elements of $\hom_A(\bc_*(M), \bc_*(N))$ ``blob cochains". |
828 |
832 |
829 Recall (Theorem \ref{thm:evaluation}) that chains on the space of mapping cylinders also act on the |
833 Recall (Theorem \ref{thm:evaluation}) that chains on the space of mapping cylinders also act on the |
830 blob complex. |
834 blob complex. |
831 An $n$-dimensional surgery cylinder is |
835 An $n$-dimensional surgery cylinder is |
832 defined to be a sequence of mapping cylinders and surgeries (Figure \ref{delfig2}), |
836 defined to be a sequence of mapping cylinders and surgeries (Figure \ref{delfig2}), |
834 One can associated to this data an $(n{+}1)$-manifold with a foliation by intervals, |
838 One can associated to this data an $(n{+}1)$-manifold with a foliation by intervals, |
835 and the relations we impose correspond to homeomorphisms of the $(n{+}1)$-manifolds |
839 and the relations we impose correspond to homeomorphisms of the $(n{+}1)$-manifolds |
836 which preserve the foliation. |
840 which preserve the foliation. |
837 |
841 |
838 Surgery cylinders form an operad, by gluing the outer boundary of one cylinder into an inner boundary of another. |
842 Surgery cylinders form an operad, by gluing the outer boundary of one cylinder into an inner boundary of another. |
839 \nn{more to do...} |
|
840 |
843 |
841 \begin{thm}[Higher dimensional Deligne conjecture] |
844 \begin{thm}[Higher dimensional Deligne conjecture] |
842 \label{thm:deligne} |
845 \label{thm:deligne} |
843 The singular chains of the $n$-dimensional surgery cylinder operad act on blob cochains. |
846 The singular chains of the $n$-dimensional surgery cylinder operad act on blob cochains. |
844 Since the little $n{+}1$-balls operad is a suboperad of the $n$-SC operad, |
|
845 this implies that the little $n{+}1$-balls operad acts on blob cochains of the $n$-ball. |
|
846 \end{thm} |
847 \end{thm} |
847 |
848 |
848 By the `blob cochains' of a manifold $X$, we mean the $A_\infty$ maps of $\bc_*(X)$ as a $\bc_*(\bdy X)$ $A_\infty$-module. |
849 More specifically, let $M_0, N_0, \ldots, M_k, N_k$ be $n$-manifolds and let $SC^n_{\overline{M}, \overline{N}}$ |
|
850 denote the component of the operad with outer boundary $M_0\cup N_0$ and inner boundaries |
|
851 $M_1\cup N_1,\ldots, M_k\cup N_k$. |
|
852 Then there is a collection of chain maps |
|
853 \begin{multline*} |
|
854 C_*(SC^n_{\overline{M}, \overline{N}})\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots \\ |
|
855 \otimes \hom(\bc_*(M_{k}), \bc_*(N_{k})) \to \hom(\bc_*(M_0), \bc_*(N_0)) |
|
856 \end{multline*} |
|
857 which satisfy the operad compatibility conditions. |
849 |
858 |
850 \begin{proof} |
859 \begin{proof} |
851 We have already defined the action of mapping cylinders, in Theorem \ref{thm:evaluation}, |
860 We have already defined the action of mapping cylinders, in Theorem \ref{thm:evaluation}, |
852 and the action of surgeries is just composition of maps of $A_\infty$-modules. |
861 and the action of surgeries is just composition of maps of $A_\infty$-modules. |
853 We only need to check that the relations of the $n$-SC operad are satisfied. |
862 We only need to check that the relations of the $n$-SC operad are satisfied. |
854 This follows from the locality of the action of $\CH{-}$ (i.e., that it is compatible with gluing) and associativity. |
863 This follows from the locality of the action of $\CH{-}$ (i.e., that it is compatible with gluing) and associativity. |
855 \end{proof} |
864 \end{proof} |
856 |
865 |
857 The little disks operad $LD$ is homotopy equivalent to |
866 Consider the special case where $n=1$ and all of the $M_i$'s and $N_i$'s are 1-balls. |
858 \nn{suboperad of} |
867 We have that $SC^1_{\overline{M}, \overline{N}}$ is homotopy equivalent to the little |
859 the $n=1$ case of the $n$-SC operad. The blob complex $\bc_*(I, \cC)$ is a bimodule over itself, and the $A_\infty$-bimodule intertwiners are homotopy equivalent to the Hochschild cochains $Hoch^*(C, C)$. |
868 disks operad and $\hom(\bc_*(M_i), \bc_*(N_i))$ is homotopy equivalent to Hochschild cochains. |
860 The usual Deligne conjecture (proved variously in \cite{hep-th/9403055, MR1805894, MR2064592, MR1805923}) |
869 This special case is just the usual Deligne conjecture |
|
870 (see \cite{hep-th/9403055, MR1805894, MR2064592, MR1805923} |
861 \nn{should check that this is the optimal list of references; what about Gerstenhaber-Voronov?; |
871 \nn{should check that this is the optimal list of references; what about Gerstenhaber-Voronov?; |
862 if we revise this list, should propagate change back to main paper} |
872 if we revise this list, should propagate change back to main paper} |
863 gives a map |
873 ). |
864 \[ |
874 |
865 C_*(LD_k)\tensor \overbrace{Hoch^*(C, C)\tensor\cdots\tensor Hoch^*(C, C)}^{\text{$k$ copies}} |
875 The general case when $n=1$ goes beyond the original Deligne conjecture, as the $M_i$'s and $N_i$'s |
866 \to Hoch^*(C, C), |
876 could be disjoint unions of 1-balls and circles, and the surgery cylinders could be high genus surfaces. |
867 \] |
877 |
868 which we now see to be a specialization of Theorem \ref{thm:deligne}. |
878 If all of the $M_i$'s and $N_i$'s are $n$-balls, then $SC^n_{\overline{M}, \overline{N}}$ |
|
879 contains a copy of the little $(n{+}1)$-balls operad. |
|
880 Thus the little $(n{+}1)$-balls operad acts on blob cochains of the $n$-ball. |
|
881 |
869 |
882 |
870 |
883 |
871 %% == end of paper: |
884 %% == end of paper: |
872 |
885 |
873 %% Optional Materials and Methods Section |
886 %% Optional Materials and Methods Section |