679 \item $F_*(M_1 \oplus M_2) \cong F_*(M_1) \oplus F_*(M_2)$. |
680 \item $F_*(M_1 \oplus M_2) \cong F_*(M_1) \oplus F_*(M_2)$. |
680 \item An exact sequence $0 \to M_1 \to M_2 \to M_3 \to 0$ |
681 \item An exact sequence $0 \to M_1 \to M_2 \to M_3 \to 0$ |
681 gives rise to an exact sequence $0 \to F_*(M_1) \to F_*(M_2) \to F_*(M_3) \to 0$. |
682 gives rise to an exact sequence $0 \to F_*(M_1) \to F_*(M_2) \to F_*(M_3) \to 0$. |
682 (See below for proof.) |
683 (See below for proof.) |
683 \item $F_*(C\otimes C)$ (the free $C$-$C$-bimodule with one generator) is |
684 \item $F_*(C\otimes C)$ (the free $C$-$C$-bimodule with one generator) is |
684 homotopic to the 0-step complex $C$. |
685 quasi-isomorphic to the 0-step complex $C$. |
685 (See below for proof.) |
686 (See below for proof.) |
686 \item $F_*(C)$ (here $C$ is wearing its $C$-$C$-bimodule hat) is homotopic to $\bc_*(S^1)$. |
687 \item $F_*(C)$ (here $C$ is wearing its $C$-$C$-bimodule hat) is quasi-isomorphic to $\bc_*(S^1)$. |
687 (See below for proof.) |
688 (See below for proof.) |
688 \end{itemize} |
689 \end{itemize} |
689 |
690 |
690 First we show that $F_*(C\otimes C)$ is |
691 First we show that $F_*(C\otimes C)$ is |
691 homotopic to the 0-step complex $C$. |
692 quasi-isomorphic to the 0-step complex $C$. |
692 |
693 |
693 Let $F'_* \sub F_*(C\otimes C)$ be the subcomplex where the label of |
694 Let $F'_* \sub F_*(C\otimes C)$ be the subcomplex where the label of |
694 the point $*$ is $1 \otimes 1 \in C\otimes C$. |
695 the point $*$ is $1 \otimes 1 \in C\otimes C$. |
695 We will show that the inclusion $i: F'_* \to F_*(C\otimes C)$ is a quasi-isomorphism. |
696 We will show that the inclusion $i: F'_* \to F_*(C\otimes C)$ is a quasi-isomorphism. |
696 |
697 |
697 Fix a small $\ep > 0$. |
698 Fix a small $\ep > 0$. |
698 Let $B_\ep$ be the ball of radius $\ep$ around $* \in S^1$. |
699 Let $B_\ep$ be the ball of radius $\ep$ around $* \in S^1$. |
699 Let $F^\ep_* \sub F_*(C\otimes C)$ be the subcomplex where $B_\ep$ is either disjoint from |
700 Let $F^\ep_* \sub F_*(C\otimes C)$ be the subcomplex |
700 or contained in all blobs, and the two boundary points of $B_\ep$ are not labeled points. |
701 generated by blob diagrams $b$ such that $B_\ep$ is either disjoint from |
|
702 or contained in each blob of $b$, and the two boundary points of $B_\ep$ are not labeled points of $b$. |
701 For a field (picture) $y$ on $B_\ep$, let $s_\ep(y)$ be the equivalent picture with~$*$ |
703 For a field (picture) $y$ on $B_\ep$, let $s_\ep(y)$ be the equivalent picture with~$*$ |
702 labeled by $1\otimes 1$ and the only other labeled points at distance $\pm\ep/2$ from $*$. |
704 labeled by $1\otimes 1$ and the only other labeled points at distance $\pm\ep/2$ from $*$. |
703 (See Figure xxxx.) |
705 (See Figure xxxx.) |
|
706 Note that $y - s_\ep(y) \in U(B_\ep)$. |
704 \nn{maybe it's simpler to assume that there are no labeled points, other than $*$, in $B_\ep$.} |
707 \nn{maybe it's simpler to assume that there are no labeled points, other than $*$, in $B_\ep$.} |
705 |
708 |
706 Define a degree 1 chain map $j_\ep : F^\ep_* \to F^\ep_*$ as follows. |
709 Define a degree 1 chain map $j_\ep : F^\ep_* \to F^\ep_*$ as follows. |
707 Let $x \in F^\ep_*$ be a blob diagram. |
710 Let $x \in F^\ep_*$ be a blob diagram. |
708 If $*$ is not contained in any twig blob, $j_\ep(x)$ is obtained by adding $B_\ep$ to |
711 If $*$ is not contained in any twig blob, $j_\ep(x)$ is obtained by adding $B_\ep$ to |
709 $x$ as a new twig blob, with label $y - s_\ep(y)$, where $y$ is the restriction of $x$ to $B_\ep$. |
712 $x$ as a new twig blob, with label $y - s_\ep(y)$, where $y$ is the restriction of $x$ to $B_\ep$. |
710 If $*$ is contained in a twig blob $B$ with label $u = \sum z_i$, $j_\ep(x)$ is obtained as follows. |
713 If $*$ is contained in a twig blob $B$ with label $u = \sum z_i$, $j_\ep(x)$ is obtained as follows. |
711 Let $y_i$ be the restriction of $z_i$ to $*$. |
714 Let $y_i$ be the restriction of $z_i$ to $B_\ep$. |
712 Let $x_i$ be equal to $x$ outside of $B$, equal to $z_i$ on $B \setmin B_\ep$, |
715 Let $x_i$ be equal to $x$ outside of $B$, equal to $z_i$ on $B \setmin B_\ep$, |
713 and have an additional blob $B_\ep$ with label $y_i - s_\ep(y_i)$. |
716 and have an additional blob $B_\ep$ with label $y_i - s_\ep(y_i)$. |
714 Define $j_\ep(x) = \sum x_i$. |
717 Define $j_\ep(x) = \sum x_i$. |
|
718 \nn{need to check signs coming from blob complex differential} |
715 |
719 |
716 Note that if $x \in F'_* \cap F^\ep_*$ then $j_\ep(x) \in F'_*$ also. |
720 Note that if $x \in F'_* \cap F^\ep_*$ then $j_\ep(x) \in F'_*$ also. |
717 |
721 |
718 The key property of $j_\ep$ is |
722 The key property of $j_\ep$ is |
719 \eq{ |
723 \eq{ |
720 \bd j_\ep + j_\ep \bd = \id - \sigma_\ep , |
724 \bd j_\ep + j_\ep \bd = \id - \sigma_\ep , |
721 } |
725 } |
722 where $\sigma_\ep : F^\ep_* \to F^\ep_*$ is given by replacing the restriction of each field |
726 where $\sigma_\ep : F^\ep_* \to F^\ep_*$ is given by replacing the restriction $y$ of each field |
723 mentioned in $x \in F^\ep_*$ (call the restriction $y$) with $s_\ep(y)$. |
727 mentioned in $x \in F^\ep_*$ with $s_\ep(y)$. |
724 Note that $\sigma_\ep(x) \in F'$. |
728 Note that $\sigma_\ep(x) \in F'_*$. |
725 |
729 |
726 If $j_\ep$ were defined on all of $F_*(C\otimes C)$, it would show that $\sigma_\ep$ |
730 If $j_\ep$ were defined on all of $F_*(C\otimes C)$, it would show that $\sigma_\ep$ |
727 is a homotopy inverse to the inclusion $F'_* \to F_*(C\otimes C)$. |
731 is a homotopy inverse to the inclusion $F'_* \to F_*(C\otimes C)$. |
728 One strategy would be to try to stitch together various $j_\ep$ for progressively smaller |
732 One strategy would be to try to stitch together various $j_\ep$ for progressively smaller |
729 $\ep$ and show that $F'_*$ is homotopy equivalent to $F_*(C\otimes C)$. |
733 $\ep$ and show that $F'_*$ is homotopy equivalent to $F_*(C\otimes C)$. |
730 Instead, we'll be less ambitious and just show that |
734 Instead, we'll be less ambitious and just show that |
731 $F'_*$ is quasi-isomorphic to $F_*(C\otimes C)$. |
735 $F'_*$ is quasi-isomorphic to $F_*(C\otimes C)$. |
732 |
736 |
733 If $x$ is a cycle in $F_*(C\otimes C)$, then for sufficiently small $\ep$ |
737 If $x$ is a cycle in $F_*(C\otimes C)$, then for sufficiently small $\ep$ we have |
734 $x \in F_*^\ep$. |
738 $x \in F_*^\ep$. |
735 (This is true for any chain in $F_*(C\otimes C)$, since chains are sums of |
739 (This is true for any chain in $F_*(C\otimes C)$, since chains are sums of |
736 finitely many blob diagrams.) |
740 finitely many blob diagrams.) |
737 Then $x$ is homologous to $s_\ep(x)$, which is in $F'_*$, so the inclusion map |
741 Then $x$ is homologous to $s_\ep(x)$, which is in $F'_*$, so the inclusion map |
738 is surjective on homology. |
742 $F'_* \sub F_*(C\otimes C)$ is surjective on homology. |
739 If $y \in F_*(C\otimes C)$ and $\bd y = x \in F'_*$, then $y \in F^\ep_*$ for some $\ep$ |
743 If $y \in F_*(C\otimes C)$ and $\bd y = x \in F'_*$, then $y \in F^\ep_*$ for some $\ep$ |
740 and |
744 and |
741 \eq{ |
745 \eq{ |
742 \bd x = \bd (\sigma_\ep(y) + j_\ep(x)) . |
746 \bd y = \bd (\sigma_\ep(y) + j_\ep(x)) . |
743 } |
747 } |
744 Since $\sigma_\ep(y) + j_\ep(x) \in F'$, it follows that the inclusion map is injective on homology. |
748 Since $\sigma_\ep(y) + j_\ep(x) \in F'$, it follows that the inclusion map is injective on homology. |
745 This completes the proof that $F'_*$ is quasi-isomorphic to $F_*(C\otimes C)$. |
749 This completes the proof that $F'_*$ is quasi-isomorphic to $F_*(C\otimes C)$. |
746 |
750 |
747 \medskip |
751 \medskip |
749 Let $F''_* \sub F'_*$ be the subcomplex of $F'_*$ where $*$ is not contained in any blob. |
753 Let $F''_* \sub F'_*$ be the subcomplex of $F'_*$ where $*$ is not contained in any blob. |
750 We will show that the inclusion $i: F''_* \to F'_*$ is a homotopy equivalence. |
754 We will show that the inclusion $i: F''_* \to F'_*$ is a homotopy equivalence. |
751 |
755 |
752 First, a lemma: Let $G''_*$ and $G'_*$ be defined the same as $F''_*$ and $F'_*$, except with |
756 First, a lemma: Let $G''_*$ and $G'_*$ be defined the same as $F''_*$ and $F'_*$, except with |
753 $S^1$ replaced some (any) neighborhood of $* \in S^1$. |
757 $S^1$ replaced some (any) neighborhood of $* \in S^1$. |
754 Then $G''_*$ and $G'_*$ are both contractible. |
758 Then $G''_*$ and $G'_*$ are both contractible |
|
759 and the inclusion $G''_* \sub G'_*$ is a homotopy equivalence. |
755 For $G'_*$ the proof is the same as in (\ref{bcontract}), except that the splitting |
760 For $G'_*$ the proof is the same as in (\ref{bcontract}), except that the splitting |
756 $G'_0 \to H_0(G'_*)$ concentrates the point labels at two points to the right and left of $*$. |
761 $G'_0 \to H_0(G'_*)$ concentrates the point labels at two points to the right and left of $*$. |
757 For $G''_*$ we note that any cycle is supported \nn{need to establish terminology for this; maybe |
762 For $G''_*$ we note that any cycle is supported \nn{need to establish terminology for this; maybe |
758 in ``basic properties" section above} away from $*$. |
763 in ``basic properties" section above} away from $*$. |
759 Thus any cycle lies in the image of the normal blob complex of a disjoint union |
764 Thus any cycle lies in the image of the normal blob complex of a disjoint union |
796 This completes the proof that $F_*(C\otimes C)$ is |
801 This completes the proof that $F_*(C\otimes C)$ is |
797 homotopic to the 0-step complex $C$. |
802 homotopic to the 0-step complex $C$. |
798 |
803 |
799 \medskip |
804 \medskip |
800 |
805 |
801 Next we show that $F_*(C)$ is homotopic \nn{q-isom?} to $\bc_*(S^1)$ |
806 Next we show that $F_*(C)$ is quasi-isomorphic to $\bc_*(S^1)$. |
802 \nn{...} |
807 $F_*(C)$ differs from $\bc_*(S^1)$ only in that the base point * |
|
808 is always a labeled point in $F_*(C)$, while in $\bc_*(S^1)$ it may or may not be. |
|
809 In other words, there is an inclusion map $i: F_*(C) \to \bc_*(S^1)$. |
|
810 |
|
811 We define a quasi-inverse \nn{right term?} $s: \bc_*(S^1) \to F_*(C)$ to the inclusion as follows. |
|
812 If $y$ is a field defined on a neighborhood of *, define $s(y) = y$ if |
|
813 * is a labeled point in $y$. |
|
814 Otherwise, define $s(y)$ to be the result of adding a label 1 (identity morphism) at *. |
|
815 Let $x \in \bc_*(S^1)$. |
|
816 Let $s(x)$ be the result of replacing each field $y$ (containing *) mentioned in |
|
817 $x$ with $y$. |
|
818 It is easy to check that $s$ is a chain map and $s \circ i = \id$. |
|
819 |
|
820 Let $G^\ep_* \sub \bc_*(S^1)$ be the subcomplex where there are no labeled points |
|
821 in a neighborhood $B_\ep$ of *, except perhaps *. |
|
822 Note that for any chain $x \in \bc_*(S^1)$, $x \in G^\ep_*$ for sufficiently small $\ep$. |
|
823 \nn{rest of argument goes similarly to above} |
803 |
824 |
804 \bigskip |
825 \bigskip |
805 |
826 |
806 \nn{still need to prove exactness claim} |
827 \nn{still need to prove exactness claim} |
807 |
828 |
808 \nn{What else needs to be said to establish quasi-isomorphism to Hochschild complex? |
829 \nn{What else needs to be said to establish quasi-isomorphism to Hochschild complex? |
809 Do we need a map from hoch to blob? |
830 Do we need a map from hoch to blob? |
810 Does the above exactness and contractibility guarantee such a map without writing it |
831 Does the above exactness and contractibility guarantee such a map without writing it |
811 down explicitly? |
832 down explicitly? |
812 Probably it's worth writing down an explicit map even if we don't need to.} |
833 Probably it's worth writing down an explicit map even if we don't need to.} |
|
834 |
|
835 |
|
836 |
813 |
837 |
814 |
838 |
815 |
839 |
816 \section{Action of $C_*(\Diff(X))$} \label{diffsect} |
840 \section{Action of $C_*(\Diff(X))$} \label{diffsect} |
817 |
841 |
863 \begin{itemize} |
887 \begin{itemize} |
864 \item each $f_i(p, \cdot): X \to X$ is supported on some connected $V_i \sub X$; |
888 \item each $f_i(p, \cdot): X \to X$ is supported on some connected $V_i \sub X$; |
865 \item the $V_i$'s are mutually disjoint; |
889 \item the $V_i$'s are mutually disjoint; |
866 \item each $V_i$ is the union of at most $k_i$ of the $U_\alpha$'s, |
890 \item each $V_i$ is the union of at most $k_i$ of the $U_\alpha$'s, |
867 where $k_i = \dim(P_i)$; and |
891 where $k_i = \dim(P_i)$; and |
868 \item $f(p, \cdot) = f_1(p_1, \cdot) \circ \cdots \circ f_m(p_m, \cdot)$ |
892 \item $f(p, \cdot) = f_1(p_1, \cdot) \circ \cdots \circ f_m(p_m, \cdot) \circ g$ |
869 for all $p = (p_1, \ldots, p_m)$. |
893 for all $p = (p_1, \ldots, p_m)$, for some fixed $g \in \Diff(X)$. |
870 \end{itemize} |
894 \end{itemize} |
871 A chain $x \in C_k(\Diff(M))$ is (by definition) adapted to $\cU$ if is is the sum |
895 A chain $x \in C_k(\Diff(X))$ is (by definition) adapted to $\cU$ if it is the sum |
872 of singular cells, each of which is adapted to $\cU$. |
896 of singular cells, each of which is adapted to $\cU$. |
873 |
897 |
874 \begin{lemma} \label{extension_lemma} |
898 \begin{lemma} \label{extension_lemma} |
875 Let $x \in CD_k(X)$ be a singular chain such that $\bd x$ is adapted to $\cU$. |
899 Let $x \in CD_k(X)$ be a singular chain such that $\bd x$ is adapted to $\cU$. |
876 Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$. |
900 Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$. |
1024 Next we verify that $u$ has the desired properties. |
1055 Next we verify that $u$ has the desired properties. |
1025 |
1056 |
1026 Since $u(0, p, x) = p$ for all $p\in P$ and $x\in X$, $F(0, p, x) = f(p, x)$ for all $p$ and $x$. |
1057 Since $u(0, p, x) = p$ for all $p\in P$ and $x\in X$, $F(0, p, x) = f(p, x)$ for all $p$ and $x$. |
1027 Therefore $F$ is a homotopy from $f$ to something. |
1058 Therefore $F$ is a homotopy from $f$ to something. |
1028 |
1059 |
1029 Next we show that the $K_\alpha$'s are sufficiently fine cell decompositions, |
1060 Next we show that if the $K_\alpha$'s are sufficiently fine cell decompositions, |
1030 then $F$ is a homotopy through diffeomorphisms. |
1061 then $F$ is a homotopy through diffeomorphisms. |
1031 We must show that the derivative $\pd{F}{x}(t, p, x)$ is non-singular for all $(t, p, x)$. |
1062 We must show that the derivative $\pd{F}{x}(t, p, x)$ is non-singular for all $(t, p, x)$. |
1032 We have |
1063 We have |
1033 \eq{ |
1064 \eq{ |
1034 % \pd{F}{x}(t, p, x) = \pd{f}{x}(u(t, p, x), x) + \pd{f}{p}(u(t, p, x), x) \pd{u}{x}(t, p, x) . |
1065 % \pd{F}{x}(t, p, x) = \pd{f}{x}(u(t, p, x), x) + \pd{f}{p}(u(t, p, x), x) \pd{u}{x}(t, p, x) . |
1101 |
1133 |
1102 |
1134 |
1103 |
1135 |
1104 \section{$A_\infty$ action on the boundary} |
1136 \section{$A_\infty$ action on the boundary} |
1105 |
1137 |
|
1138 Let $Y$ be an $n{-}1$-manifold. |
|
1139 The collection of complexes $\{\bc_*(Y\times I; a, b)\}$, where $a, b \in \cC(Y)$ are boundary |
|
1140 conditions on $\bd(Y\times I) = Y\times \{0\} \cup Y\times\{1\}$, has the structure |
|
1141 of an $A_\infty$ category. |
|
1142 |
|
1143 Composition of morphisms (multiplication) depends of a choice of homeomorphism |
|
1144 $I\cup I \cong I$. Given this choice, gluing gives a map |
|
1145 \eq{ |
|
1146 \bc_*(Y\times I; a, b) \otimes \bc_*(Y\times I; b, c) \to \bc_*(Y\times (I\cup I); a, c) |
|
1147 \cong \bc_*(Y\times I; a, c) |
|
1148 } |
|
1149 Using (\ref{CDprop}) and the inclusion $\Diff(I) \sub \Diff(Y\times I)$ gives the various |
|
1150 higher associators of the $A_\infty$ structure, more or less canonically. |
|
1151 |
|
1152 \nn{is this obvious? does more need to be said?} |
|
1153 |
|
1154 Let $\cA(Y)$ denote the $A_\infty$ category $\bc_*(Y\times I; \cdot, \cdot)$. |
|
1155 |
|
1156 Similarly, if $Y \sub \bd X$, a choice of collaring homeomorphism |
|
1157 $(Y\times I) \cup_Y X \cong X$ gives the collection of complexes $\bc_*(X; r, a)$ |
|
1158 (variable $a \in \cC(Y)$; fixed $r \in \cC(\bd X \setmin Y)$) the structure of a representation of the |
|
1159 $A_\infty$ category $\{\bc_*(Y\times I; \cdot, \cdot)\}$. |
|
1160 Again the higher associators come from the action of $\Diff(I)$ on a collar neighborhood |
|
1161 of $Y$ in $X$. |
|
1162 |
|
1163 In the next section we use the above $A_\infty$ actions to state and prove |
|
1164 a gluing theorem for the blob complexes of $n$-manifolds. |
|
1165 |
|
1166 |
|
1167 |
|
1168 |
|
1169 |
|
1170 |
1106 |
1171 |
1107 \section{Gluing} \label{gluesect} |
1172 \section{Gluing} \label{gluesect} |
1108 |
1173 |
|
1174 Let $Y$ be an $n{-}1$-manifold and let $X$ be an $n$-manifold with a copy |
|
1175 of $Y \du -Y$ contained in its boundary. |
|
1176 Gluing the two copies of $Y$ together we obtain a new $n$-manifold $X\sgl$. |
|
1177 We wish to describe the blob complex of $X\sgl$ in terms of the blob complex |
|
1178 of $X$. |
|
1179 More precisely, we want to describe $\bc_*(X\sgl; c\sgl)$, |
|
1180 where $c\sgl \in \cC(\bd X\sgl)$, |
|
1181 in terms of the collection $\{\bc_*(X; c, \cdot, \cdot)\}$, thought of as a representation |
|
1182 of the $A_\infty$ category $\cA(Y\du-Y) \cong \cA(Y)\times \cA(Y)\op$. |
|
1183 |
|
1184 \begin{thm} |
|
1185 $\bc_*(X\sgl; c\sgl)$ is quasi-isomorphic to the the self tensor product |
|
1186 of $\{\bc_*(X; c, \cdot, \cdot)\}$ over $\cA(Y)$. |
|
1187 \end{thm} |
|
1188 |
|
1189 The proof will occupy the remainder of this section. |
|
1190 |
|
1191 \nn{...} |
|
1192 |
|
1193 \bigskip |
|
1194 |
|
1195 \nn{need to define/recall def of (self) tensor product over an $A_\infty$ category} |
|
1196 |
|
1197 |
|
1198 |
|
1199 |
|
1200 |
1109 \section{Extension to ...} |
1201 \section{Extension to ...} |
1110 |
1202 |
1111 (Need to let the input $n$-category $C$ be a graded thing |
1203 \nn{Need to let the input $n$-category $C$ be a graded thing |
1112 (e.g.~DGA or $A_\infty$ $n$-category).) |
1204 (e.g.~DGA or $A_\infty$ $n$-category).} |
|
1205 |
|
1206 \nn{maybe this should be done earlier in the exposition? |
|
1207 if we can plausibly claim that the various proofs work almost |
|
1208 the same with the extended def, then maybe it's better to extend late (here)} |
1113 |
1209 |
1114 |
1210 |
1115 \section{What else?...} |
1211 \section{What else?...} |
1116 |
1212 |
1117 \begin{itemize} |
1213 \begin{itemize} |
1118 \item Derive Hochschild standard results from blob point of view? |
1214 \item Derive Hochschild standard results from blob point of view? |
1119 \item $n=2$ examples |
1215 \item $n=2$ examples |
1120 \item Kh |
1216 \item Kh |
1121 \item dimension $n+1$ |
1217 \item dimension $n+1$ (generalized Deligne conjecture?) |
1122 \item should be clear about PL vs Diff; probably PL is better |
1218 \item should be clear about PL vs Diff; probably PL is better |
1123 (or maybe not) |
1219 (or maybe not) |
1124 \item say what we mean by $n$-category, $A_\infty$ or $E_\infty$ $n$-category |
1220 \item say what we mean by $n$-category, $A_\infty$ or $E_\infty$ $n$-category |
1125 \item something about higher derived coend things (derived 2-coend, e.g.) |
1221 \item something about higher derived coend things (derived 2-coend, e.g.) |
1126 \end{itemize} |
1222 \end{itemize} |
1127 |
1223 |
1128 |
1224 |
1129 |
1225 |
|
1226 |
|
1227 |
1130 \end{document} |
1228 \end{document} |
1131 |
1229 |
1132 |
1230 |
1133 |
1231 |
1134 %Recall that for $n$-category picture fields there is an evaluation map |
1232 %Recall that for $n$-category picture fields there is an evaluation map |