50 \def\calc#1{\expandafter\def\csname c#1\endcsname{{\mathcal #1}}} |
50 \def\calc#1{\expandafter\def\csname c#1\endcsname{{\mathcal #1}}} |
51 \applytolist{calc}QWERTYUIOPLKJHGFDSAZXCVBNM; |
51 \applytolist{calc}QWERTYUIOPLKJHGFDSAZXCVBNM; |
52 |
52 |
53 % \DeclareMathOperator{\pr}{pr} etc. |
53 % \DeclareMathOperator{\pr}{pr} etc. |
54 \def\declaremathop#1{\expandafter\DeclareMathOperator\csname #1\endcsname{#1}} |
54 \def\declaremathop#1{\expandafter\DeclareMathOperator\csname #1\endcsname{#1}} |
55 \applytolist{declaremathop}{pr}{im}{gl}{ev}{coinv}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{Tet}{cat}{Maps}{Diff}{sign}{supp}{maps}; |
55 \applytolist{declaremathop}{pr}{im}{gl}{ev}{coinv}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{Tet}{cat}{Maps}{Diff}{sign}{supp}; |
56 |
56 |
57 |
57 |
58 |
58 |
59 %%%%%% end excerpt |
59 %%%%%% end excerpt |
60 |
60 |
762 blob diagram $(b_1, b_2)$ on $X \du Y$. |
762 blob diagram $(b_1, b_2)$ on $X \du Y$. |
763 Because of the blob reordering relations, all blob diagrams on $X \du Y$ arise this way. |
763 Because of the blob reordering relations, all blob diagrams on $X \du Y$ arise this way. |
764 In the other direction, any blob diagram on $X\du Y$ is equal (up to sign) |
764 In the other direction, any blob diagram on $X\du Y$ is equal (up to sign) |
765 to one that puts $X$ blobs before $Y$ blobs in the ordering, and so determines |
765 to one that puts $X$ blobs before $Y$ blobs in the ordering, and so determines |
766 a pair of blob diagrams on $X$ and $Y$. |
766 a pair of blob diagrams on $X$ and $Y$. |
767 These two maps are compatible with our sign conventions \nn{say more about this?} and |
767 These two maps are compatible with our sign conventions. |
768 with the linear label relations. |
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769 The two maps are inverses of each other. |
768 The two maps are inverses of each other. |
770 \nn{should probably say something about sign conventions for the differential |
769 \nn{should probably say something about sign conventions for the differential |
771 in a tensor product of chain complexes; ask Scott} |
770 in a tensor product of chain complexes; ask Scott} |
772 \end{proof} |
771 \end{proof} |
773 |
772 |
776 |
775 |
777 Suppose that for all $c \in \cC(\bd B^n)$ |
776 Suppose that for all $c \in \cC(\bd B^n)$ |
778 we have a splitting $s: H_0(\bc_*(B^n, c)) \to \bc_0(B^n; c)$ |
777 we have a splitting $s: H_0(\bc_*(B^n, c)) \to \bc_0(B^n; c)$ |
779 of the quotient map |
778 of the quotient map |
780 $p: \bc_0(B^n; c) \to H_0(\bc_*(B^n, c))$. |
779 $p: \bc_0(B^n; c) \to H_0(\bc_*(B^n, c))$. |
781 \nn{always the case if we're working over $\c$}. |
780 For example, this is always the case if you coefficient ring is a field. |
782 Then |
781 Then |
783 \begin{prop} \label{bcontract} |
782 \begin{prop} \label{bcontract} |
784 For all $c \in \cC(\bd B^n)$ the natural map $p: \bc_*(B^n, c) \to H_0(\bc_*(B^n, c))$ |
783 For all $c \in \cC(\bd B^n)$ the natural map $p: \bc_*(B^n, c) \to H_0(\bc_*(B^n, c))$ |
785 is a chain homotopy equivalence |
784 is a chain homotopy equivalence |
786 with inverse $s: H_0(\bc_*(B^n, c)) \to \bc_*(B^n; c)$. |
785 with inverse $s: H_0(\bc_*(B^n, c)) \to \bc_*(B^n; c)$. |
792 For $i \ge 1$, define $h_i : \bc_i(B^n; c) \to \bc_{i+1}(B^n; c)$ by adding |
791 For $i \ge 1$, define $h_i : \bc_i(B^n; c) \to \bc_{i+1}(B^n; c)$ by adding |
793 an $(i{+}1)$-st blob equal to all of $B^n$. |
792 an $(i{+}1)$-st blob equal to all of $B^n$. |
794 In other words, add a new outermost blob which encloses all of the others. |
793 In other words, add a new outermost blob which encloses all of the others. |
795 Define $h_0 : \bc_0(B^n; c) \to \bc_1(B^n; c)$ by setting $h_0(x)$ equal to |
794 Define $h_0 : \bc_0(B^n; c) \to \bc_1(B^n; c)$ by setting $h_0(x)$ equal to |
796 the 1-blob with blob $B^n$ and label $x - s(p(x)) \in U(B^n; c)$. |
795 the 1-blob with blob $B^n$ and label $x - s(p(x)) \in U(B^n; c)$. |
797 \nn{$x$ is a 0-blob diagram, i.e. $x \in \cC(B^n; c)$} |
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798 \end{proof} |
796 \end{proof} |
799 |
797 |
800 (Note that for the above proof to work, we need the linear label relations |
798 Note that if there is no splitting $s$, we can let $h_0 = 0$ and get a homotopy |
801 for blob labels. |
799 equivalence to the 2-step complex $U(B^n; c) \to \cC(B^n; c)$. |
802 Also we need to blob reordering relations (?).) |
800 |
803 |
801 For fields based on $n$-categories, $H_0(\bc_*(B^n; c)) \cong \mor(c', c'')$, |
804 (Note also that if there is no splitting $s$, we can let $h_0 = 0$ and get a homotopy |
802 where $(c', c'')$ is some (any) splitting of $c$ into domain and range. |
805 equivalence to the 2-step complex $U(B^n; c) \to \cC(B^n; c)$.) |
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806 |
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807 (For fields based on $n$-cats, $H_0(\bc_*(B^n; c)) \cong \mor(c', c'')$.) |
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808 |
803 |
809 \medskip |
804 \medskip |
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805 |
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806 \nn{Maybe there is no longer a need to repeat the next couple of props here, since we also state them in the introduction. |
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807 But I think it's worth saying that the Diff actions will be enhanced later. |
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808 Maybe put that in the intro too.} |
810 |
809 |
811 As we noted above, |
810 As we noted above, |
812 \begin{prop} |
811 \begin{prop} |
813 There is a natural isomorphism $H_0(\bc_*(X)) \cong A(X)$. |
812 There is a natural isomorphism $H_0(\bc_*(X)) \cong A(X)$. |
814 \qed |
813 \qed |
815 \end{prop} |
814 \end{prop} |
816 |
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817 |
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818 % oops -- duplicate |
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819 |
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820 %\begin{prop} \label{functorialprop} |
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821 %The assignment $X \mapsto \bc_*(X)$ extends to a functor from the category of |
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822 %$n$-manifolds and homeomorphisms to the category of chain complexes and linear isomorphisms. |
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823 %\end{prop} |
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824 |
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825 %\begin{proof} |
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826 %Obvious. |
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827 %\end{proof} |
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828 |
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829 %\nn{need to same something about boundaries and boundary conditions above. |
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830 %maybe fix the boundary and consider the category of $n$-manifolds with the given boundary.} |
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831 |
815 |
832 |
816 |
833 \begin{prop} |
817 \begin{prop} |
834 For fixed fields ($n$-cat), $\bc_*$ is a functor from the category |
818 For fixed fields ($n$-cat), $\bc_*$ is a functor from the category |
835 of $n$-manifolds and diffeomorphisms to the category of chain complexes and |
819 of $n$-manifolds and diffeomorphisms to the category of chain complexes and |
836 (chain map) isomorphisms. |
820 (chain map) isomorphisms. |
837 \qed |
821 \qed |
838 \end{prop} |
822 \end{prop} |
839 |
823 |
840 \nn{need to same something about boundaries and boundary conditions above. |
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841 maybe fix the boundary and consider the category of $n$-manifolds with the given boundary.} |
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842 |
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843 |
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844 In particular, |
824 In particular, |
845 \begin{prop} \label{diff0prop} |
825 \begin{prop} \label{diff0prop} |
846 There is an action of $\Diff(X)$ on $\bc_*(X)$. |
826 There is an action of $\Diff(X)$ on $\bc_*(X)$. |
847 \qed |
827 \qed |
848 \end{prop} |
828 \end{prop} |
855 conditions to the notation. |
835 conditions to the notation. |
856 |
836 |
857 Let $X$ be an $n$-manifold, $\bd X = Y \cup (-Y) \cup Z$. |
837 Let $X$ be an $n$-manifold, $\bd X = Y \cup (-Y) \cup Z$. |
858 Gluing the two copies of $Y$ together yields an $n$-manifold $X\sgl$ |
838 Gluing the two copies of $Y$ together yields an $n$-manifold $X\sgl$ |
859 with boundary $Z\sgl$. |
839 with boundary $Z\sgl$. |
860 Given compatible fields (pictures, boundary conditions) $a$, $b$ and $c$ on $Y$, $-Y$ and $Z$, |
840 Given compatible fields (boundary conditions) $a$, $b$ and $c$ on $Y$, $-Y$ and $Z$, |
861 we have the blob complex $\bc_*(X; a, b, c)$. |
841 we have the blob complex $\bc_*(X; a, b, c)$. |
862 If $b = -a$ (the orientation reversal of $a$), then we can glue up blob diagrams on |
842 If $b = -a$ (the orientation reversal of $a$), then we can glue up blob diagrams on |
863 $X$ to get blob diagrams on $X\sgl$: |
843 $X$ to get blob diagrams on $X\sgl$: |
864 |
844 |
865 \begin{prop} |
845 \begin{prop} |
874 \end{prop} |
854 \end{prop} |
875 |
855 |
876 The above map is very far from being an isomorphism, even on homology. |
856 The above map is very far from being an isomorphism, even on homology. |
877 This will be fixed in Section \ref{sec:gluing} below. |
857 This will be fixed in Section \ref{sec:gluing} below. |
878 |
858 |
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859 \nn{Next para not need, since we already use bullet = gluing notation above(?)} |
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860 |
879 An instance of gluing we will encounter frequently below is where $X = X_1 \du X_2$ |
861 An instance of gluing we will encounter frequently below is where $X = X_1 \du X_2$ |
880 and $X\sgl = X_1 \cup_Y X_2$. |
862 and $X\sgl = X_1 \cup_Y X_2$. |
881 (Typically one of $X_1$ or $X_2$ is a disjoint union of balls.) |
863 (Typically one of $X_1$ or $X_2$ is a disjoint union of balls.) |
882 For $x_i \in \bc_*(X_i)$, we introduce the notation |
864 For $x_i \in \bc_*(X_i)$, we introduce the notation |
883 \eq{ |
865 \eq{ |
884 x_1 \bullet x_2 \deq \gl(x_1 \otimes x_2) . |
866 x_1 \bullet x_2 \deq \gl(x_1 \otimes x_2) . |
885 } |
867 } |
886 Note that we have resumed our habit of omitting boundary labels from the notation. |
868 Note that we have resumed our habit of omitting boundary labels from the notation. |
887 |
869 |
888 |
870 |
889 \bigskip |
871 |
890 |
872 |
891 \nn{what else?} |
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892 |
873 |
893 \section{Hochschild homology when $n=1$} |
874 \section{Hochschild homology when $n=1$} |
894 \label{sec:hochschild} |
875 \label{sec:hochschild} |
895 \input{text/hochschild} |
876 \input{text/hochschild} |
896 |
877 |