318 The action of these homeomorphisms gives the ``strong duality" structure. |
318 The action of these homeomorphisms gives the ``strong duality" structure. |
319 As such, we don't subdivide the boundary of a morphism |
319 As such, we don't subdivide the boundary of a morphism |
320 into domain and range --- the duality operations can convert between domain and range. |
320 into domain and range --- the duality operations can convert between domain and range. |
321 |
321 |
322 Later we inductively define an extension of the functors $\cC_k$ to functors $\cl{\cC}_k$ |
322 Later we inductively define an extension of the functors $\cC_k$ to functors $\cl{\cC}_k$ |
323 from arbitrary manifolds to sets. We need the restriction of these functors to $k$-spheres, |
323 from arbitrary manifolds to sets. We need these functors for $k$-spheres, |
324 for $k<n$, for the next axiom. |
324 for $k<n$, for the next axiom. |
325 |
325 |
326 \begin{axiom}[Boundaries]\label{nca-boundary} |
326 \begin{axiom}[Boundaries]\label{nca-boundary} |
327 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$. |
327 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$. |
328 These maps, for various $X$, comprise a natural transformation of functors. |
328 These maps, for various $X$, comprise a natural transformation of functors. |
381 The gluing maps above are strictly associative. |
381 The gluing maps above are strictly associative. |
382 Given any decomposition of a ball $B$ into smaller balls |
382 Given any decomposition of a ball $B$ into smaller balls |
383 $$\bigsqcup B_i \to B,$$ |
383 $$\bigsqcup B_i \to B,$$ |
384 any sequence of gluings (where all the intermediate steps are also disjoint unions of balls) yields the same result. |
384 any sequence of gluings (where all the intermediate steps are also disjoint unions of balls) yields the same result. |
385 \end{axiom} |
385 \end{axiom} |
|
386 This axiom is only reasonable because the definition assigns a set to every ball; any identifications would limit the extent to which we can demand associativity. |
386 For the next axiom, a \emph{pinched product} is a map locally modeled on a degeneracy map between simplices. |
387 For the next axiom, a \emph{pinched product} is a map locally modeled on a degeneracy map between simplices. |
387 \begin{axiom}[Product (identity) morphisms] |
388 \begin{axiom}[Product (identity) morphisms] |
388 \label{axiom:product} |
389 \label{axiom:product} |
389 For each pinched product $\pi:E\to X$, with $X$ a $k$-ball and $E$ a $k{+}m$-ball ($m\ge 1$), |
390 For each pinched product $\pi:E\to X$, with $X$ a $k$-ball and $E$ a $k{+}m$-ball ($m\ge 1$), |
390 there is a map $\pi^*:\cC(X)\to \cC(E)$. |
391 there is a map $\pi^*:\cC(X)\to \cC(E)$. |
475 \label{axiom:families} |
476 \label{axiom:families} |
476 For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes |
477 For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes |
477 \[ |
478 \[ |
478 C_*(\Homeo_\bd(X))\tensor \cC(X; c) \to \cC(X; c) . |
479 C_*(\Homeo_\bd(X))\tensor \cC(X; c) \to \cC(X; c) . |
479 \] |
480 \] |
480 These action maps are required to be associative up to homotopy, |
481 These action maps are required to restrict to the usual action of homeomorphisms on $C_0$, be associative up to homotopy, |
481 and also compatible with composition (gluing) in the sense that |
482 and also be compatible with composition (gluing) in the sense that |
482 a diagram like the one in Theorem \ref{thm:CH} commutes. |
483 a diagram like the one in Theorem \ref{thm:CH} commutes. |
483 \end{axiom} |
484 \end{axiom} |
484 |
485 |
485 \subsection{Example (the fundamental $n$-groupoid)} |
486 \subsection{Example (the fundamental $n$-groupoid)} |
486 We will define $\pi_{\le n}(T)$, the fundamental $n$-groupoid of a topological space $T$. |
487 We will define $\pi_{\le n}(T)$, the fundamental $n$-groupoid of a topological space $T$. |
510 to $\bd X$. |
511 to $\bd X$. |
511 For an $n$-ball $X$ define $\Bord^n(X)$ to be homeomorphism classes rel boundary of such $n$-dimensional submanifolds. |
512 For an $n$-ball $X$ define $\Bord^n(X)$ to be homeomorphism classes rel boundary of such $n$-dimensional submanifolds. |
512 |
513 |
513 There is an $A_\infty$ analogue enriched in topological spaces, where at the top level we take |
514 There is an $A_\infty$ analogue enriched in topological spaces, where at the top level we take |
514 all such submanifolds, rather than homeomorphism classes. |
515 all such submanifolds, rather than homeomorphism classes. |
515 For each fixed $\bdy W \subset \bdy X \times \bbR^\infty$, we can |
516 For each fixed $\bdy W \subset \bdy X \times \bbR^\infty$, we |
516 topologize the set of submanifolds by ambient isotopy rel boundary. |
517 topologize the set of submanifolds by ambient isotopy rel boundary. |
517 |
518 |
518 \subsection{The blob complex} |
519 \subsection{The blob complex} |
519 \subsubsection{Decompositions of manifolds} |
520 \subsubsection{Decompositions of manifolds} |
520 |
521 |
798 Blob diagrams have a natural topology, which is ignored by $\bc_*(X)$. |
799 Blob diagrams have a natural topology, which is ignored by $\bc_*(X)$. |
799 In $\cB\cT_*(X)$ we take this topology into account, treating the blob diagrams as something |
800 In $\cB\cT_*(X)$ we take this topology into account, treating the blob diagrams as something |
800 analogous to a simplicial space (but with cone-product polyhedra replacing simplices). |
801 analogous to a simplicial space (but with cone-product polyhedra replacing simplices). |
801 More specifically, a generator of $\cB\cT_k(X)$ is an $i$-parameter family of $j$-blob diagrams, with $i+j=k$. |
802 More specifically, a generator of $\cB\cT_k(X)$ is an $i$-parameter family of $j$-blob diagrams, with $i+j=k$. |
802 An essential step in the proof of this equivalence is a result to the effect that a $k$-parameter |
803 An essential step in the proof of this equivalence is a result to the effect that a $k$-parameter |
803 family of homeomorphism can be localized to at most $k$ small sets. |
804 family of homeomorphisms can be localized to at most $k$ small sets. |
804 |
805 |
805 With this alternate version in hand, it is straightforward to prove the theorem. |
806 With this alternate version in hand, the theorem is straightforward. |
806 The evaluation map $\Homeo(X)\times BD_j(X)\to BD_j(X)$ |
807 By functoriality (Property \cite{property:functoriality}) $Homeo(X)$ acts on the set $BD_j(X)$ of $j$-blob diagrams, and this |
807 induces a chain map $\CH{X}\tensor C_*(BD_j(X))\to C_*(BD_j(X))$ |
808 induces a chain map $\CH{X}\tensor C_*(BD_j(X))\to C_*(BD_j(X))$ |
808 and hence a map $e_X: \CH{X} \tensor \cB\cT_*(X) \to \cB\cT_*(X)$. |
809 and hence a map $e_X: \CH{X} \tensor \cB\cT_*(X) \to \cB\cT_*(X)$. |
809 It is easy to check that $e_X$ thus defined has the desired properties. |
810 It is easy to check that $e_X$ thus defined has the desired properties. |
810 \end{proof} |
811 \end{proof} |
811 |
812 |
829 Fix a topological $n$-category $\cC$, which we'll now omit from notation. |
830 Fix a topological $n$-category $\cC$, which we'll now omit from notation. |
830 Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ category. |
831 Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ category. |
831 |
832 |
832 \begin{thm}[Gluing formula] |
833 \begin{thm}[Gluing formula] |
833 \label{thm:gluing} |
834 \label{thm:gluing} |
834 \mbox{}% <-- gets the indenting right |
835 \mbox{}\vspace{-0.2cm}% <-- gets the indenting right |
835 \begin{itemize} |
836 \begin{itemize} |
836 \item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, |
837 \item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, |
837 the blob complex of $X$ is naturally an |
838 the blob complex of $X$ is naturally an |
838 $A_\infty$ module for $\bc_*(Y)$. |
839 $A_\infty$ module for $\bc_*(Y)$. |
839 |
840 |
850 and the $C_*(\Homeo(-))$ action of Theorem \ref{thm:evaluation}. |
851 and the $C_*(\Homeo(-))$ action of Theorem \ref{thm:evaluation}. |
851 |
852 |
852 Let $T_*$ denote the self tensor product of $\bc_*(X)$, which is a homotopy colimit. |
853 Let $T_*$ denote the self tensor product of $\bc_*(X)$, which is a homotopy colimit. |
853 There is a tautological map from the 0-simplices of $T_*$ to $\bc_*(X\bigcup_Y \selfarrow)$, |
854 There is a tautological map from the 0-simplices of $T_*$ to $\bc_*(X\bigcup_Y \selfarrow)$, |
854 and this map can be extended to a chain map on all of $T_*$ by sending the higher simplices to zero. |
855 and this map can be extended to a chain map on all of $T_*$ by sending the higher simplices to zero. |
855 Constructing a homotopy inverse to this natural map invloves making various choices, but one can show that the |
856 Constructing a homotopy inverse to this natural map involves making various choices, but one can show that the |
856 choices form contractible subcomplexes and apply the acyclic models theorem. |
857 choices form contractible subcomplexes and apply the theory of acyclic models. |
857 \end{proof} |
858 \end{proof} |
858 |
859 |
859 We next describe the blob complex for product manifolds, in terms of the $A_\infty$ |
860 We next describe the blob complex for product manifolds, in terms of the |
860 blob complex of the $A_\infty$ $n$-categories constructed as above. |
861 blob complexes for the $A_\infty$ $n$-categories constructed as above. |
861 |
862 |
862 \begin{thm}[Product formula] |
863 \begin{thm}[Product formula] |
863 \label{thm:product} |
864 \label{thm:product} |
864 Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. |
865 Let $W$ be a $k$-manifold and $Y$ be an $n{-}k$ manifold. |
865 Let $\cC$ be an $n$-category. |
866 Let $\cC$ be a linear $n$-category. |
866 Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ as above. |
867 Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ as above. |
867 Then |
868 Then |
868 \[ |
869 \[ |
869 \bc_*(Y\times W; \cC) \simeq \clh{\bc_*(Y;\cC)}(W). |
870 \bc_*(Y\times W; \cC) \simeq \clh{\bc_*(Y;\cC)}(W). |
870 \] |
871 \] |
879 diagram on $W\times Y$. |
880 diagram on $W\times Y$. |
880 This map can be extended to all of $\clh{\bc_*(Y;\cC)}(W)$ by sending higher simplices to zero. |
881 This map can be extended to all of $\clh{\bc_*(Y;\cC)}(W)$ by sending higher simplices to zero. |
881 |
882 |
882 To construct the homotopy inverse of the above map one first shows that |
883 To construct the homotopy inverse of the above map one first shows that |
883 $\bc_*(Y\times W; \cC)$ is homotopy equivalent to the subcomplex generated by blob diagrams which |
884 $\bc_*(Y\times W; \cC)$ is homotopy equivalent to the subcomplex generated by blob diagrams which |
884 are small with respect any fixed open cover of $Y\times W$. |
885 are small with respect to any fixed open cover of $Y\times W$. |
885 For a sufficiently fine open cover the generators of this ``small" blob complex are in the image of the map |
886 For a sufficiently fine open cover the generators of this ``small" blob complex are in the image of the map |
886 of the previous paragraph, and furthermore the preimage in $\clh{\bc_*(Y;\cC)}(W)$ of such small diagrams |
887 of the previous paragraph, and furthermore the preimage in $\clh{\bc_*(Y;\cC)}(W)$ of such small diagrams |
887 lie in contractible subcomplexes. |
888 lie in contractible subcomplexes. |
888 A standard acyclic models argument now constructs the homotopy inverse. |
889 A standard acyclic models argument now constructs the homotopy inverse. |
889 \end{proof} |
890 \end{proof} |
896 Let $M$ and $N$ be $n$-manifolds with common boundary $E$. |
897 Let $M$ and $N$ be $n$-manifolds with common boundary $E$. |
897 Recall (Theorem \ref{thm:gluing}) that the $A_\infty$ category $A = \bc_*(E)$ |
898 Recall (Theorem \ref{thm:gluing}) that the $A_\infty$ category $A = \bc_*(E)$ |
898 acts on $\bc_*(M)$ and $\bc_*(N)$. |
899 acts on $\bc_*(M)$ and $\bc_*(N)$. |
899 Let $\hom_A(\bc_*(M), \bc_*(N))$ denote the chain complex of $A_\infty$ module maps |
900 Let $\hom_A(\bc_*(M), \bc_*(N))$ denote the chain complex of $A_\infty$ module maps |
900 from $\bc_*(M)$ to $\bc_*(N)$. |
901 from $\bc_*(M)$ to $\bc_*(N)$. |
901 Let $R$ be another $n$-manifold with boundary $-E$. |
902 Let $R$ be another $n$-manifold with boundary $E^\text{op}$. |
902 There is a chain map |
903 There is a chain map |
903 \[ |
904 \[ |
904 \hom_A(\bc_*(M), \bc_*(N)) \ot \bc_*(M) \ot_A \bc_*(R) \to \bc_*(N) \ot_A \bc_*(R) . |
905 \hom_A(\bc_*(M), \bc_*(N)) \ot \bc_*(M) \ot_A \bc_*(R) \to \bc_*(N) \ot_A \bc_*(R) . |
905 \] |
906 \] |
906 We think of this map as being associated to a surgery which cuts $M$ out of $M\cup_E R$ and |
907 We think of this map as being associated to a surgery which cuts $M$ out of $M\cup_E R$ and |
940 and the action of surgeries is just composition of maps of $A_\infty$-modules. |
941 and the action of surgeries is just composition of maps of $A_\infty$-modules. |
941 We only need to check that the relations of the $n$-SC operad are satisfied. |
942 We only need to check that the relations of the $n$-SC operad are satisfied. |
942 This follows from the locality of the action of $\CH{-}$ (i.e., that it is compatible with gluing) and associativity. |
943 This follows from the locality of the action of $\CH{-}$ (i.e., that it is compatible with gluing) and associativity. |
943 \end{proof} |
944 \end{proof} |
944 |
945 |
945 Consider the special case where $n=1$ and all of the $M_i$'s and $N_i$'s are 1-balls. |
946 Consider the special case where $n=1$ and all of the $M_i$'s and $N_i$'s are intervals. |
946 We have that $SC^1_{\overline{M}, \overline{N}}$ is homotopy equivalent to the little |
947 We have that $SC^1_{\overline{M}, \overline{N}}$ is homotopy equivalent to the little |
947 disks operad and $\hom(\bc_*(M_i), \bc_*(N_i))$ is homotopy equivalent to Hochschild cochains. |
948 disks operad and $\hom(\bc_*(M_i), \bc_*(N_i))$ is homotopy equivalent to Hochschild cochains. |
948 This special case is just the usual Deligne conjecture |
949 This special case is just the usual Deligne conjecture |
949 (see \cite{hep-th/9403055, MR1328534, MR1805894, MR1805923, MR2064592}). |
950 (see \cite{hep-th/9403055, MR1328534, MR1805894, MR1805923, MR2064592}). |
950 |
951 |
951 The general case when $n=1$ goes beyond the original Deligne conjecture, as the $M_i$'s and $N_i$'s |
952 The general case when $n=1$ goes beyond the original Deligne conjecture, as the $M_i$'s and $N_i$'s |
952 could be disjoint unions of 1-balls and circles, and the surgery cylinders could be high genus surfaces. |
953 could be disjoint unions of intervals and circles, and the surgery cylinders could be high genus surfaces. |
953 |
954 |
954 If all of the $M_i$'s and $N_i$'s are $n$-balls, then $SC^n_{\overline{M}, \overline{N}}$ |
955 If all of the $M_i$'s and $N_i$'s are $n$-balls, then $SC^n_{\overline{M}, \overline{N}}$ |
955 contains a copy of the little $(n{+}1)$-balls operad. |
956 contains a copy of the little $(n{+}1)$-balls operad. |
956 Thus the little $(n{+}1)$-balls operad acts on blob cochains of the $n$-ball. |
957 Thus the little $(n{+}1)$-balls operad acts on blob cochains of the $n$-ball. |
957 |
958 |