90 (2) want answer independent of handle decomp (i.e. don't |
108 (2) want answer independent of handle decomp (i.e. don't |
91 just go from coend to derived coend (e.g. Hochschild homology)); |
109 just go from coend to derived coend (e.g. Hochschild homology)); |
92 (3) ... |
110 (3) ... |
93 ) |
111 ) |
94 |
112 |
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113 |
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114 |
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115 We then show that blob homology enjoys the following |
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116 \ref{property:gluing} properties. |
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117 |
|
118 \begin{property}[Functoriality] |
|
119 \label{property:functoriality}% |
|
120 Blob homology is functorial with respect to diffeomorphisms. That is, fixing an $n$-dimensional system of fields $\cF$ and local relations $\cU$, the association |
|
121 \begin{equation*} |
|
122 X \mapsto \bc_*^{\cF,\cU}(X) |
|
123 \end{equation*} |
|
124 is a functor from $n$-manifolds and diffeomorphisms between them to chain complexes and isomorphisms between them. |
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125 \scott{Do we want to or need to weaken `isomorphisms' to `homotopy equivalences' or `quasi-isomorphisms'?} |
|
126 \end{property} |
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127 |
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128 \begin{property}[Disjoint union] |
|
129 \label{property:disjoint-union} |
|
130 The blob complex of a disjoint union is naturally the tensor product of the blob complexes. |
|
131 \begin{equation*} |
|
132 \bc_*(X_1 \sqcup X_2) \iso \bc_*(X_1) \tensor \bc_*(X_2) |
|
133 \end{equation*} |
|
134 \end{property} |
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135 |
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136 \begin{property}[A map for gluing] |
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137 \label{property:gluing-map}% |
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138 If $X_1$ and $X_2$ are $n$-manifolds, with $Y$ a codimension $0$-submanifold of $\bdy X_1$, and $Y^{\text{op}}$ a codimension $0$-submanifold of $\bdy X_2$, |
|
139 there is a chain map |
|
140 \begin{equation*} |
|
141 \gl_Y: \bc_*(X_1) \tensor \bc_*(X_2) \to \bc_*(X_1 \cup_Y X_2). |
|
142 \end{equation*} |
|
143 \end{property} |
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144 |
|
145 \begin{property}[Contractibility] |
|
146 \label{property:contractibility}% |
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147 \todo{Err, requires a splitting?} |
|
148 The blob complex for an $n$-category on an $n$-ball is quasi-isomorphic to its $0$-th homology. |
|
149 \begin{equation} |
|
150 \xymatrix{\bc_*^{\cC}(B^n) \ar[r]^{\iso}_{\text{qi}} & H_0(\bc_*^{\cC}(B^n))} |
|
151 \end{equation} |
|
152 \todo{Say that this is just the original $n$-category?} |
|
153 \end{property} |
|
154 |
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155 \begin{property}[Skein modules] |
|
156 \label{property:skein-modules}% |
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157 The $0$-th blob homology of $X$ is the usual skein module associated to $X$. (See \S \ref{sec:local-relations}.) |
|
158 \begin{equation*} |
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159 H_0(\bc_*^{\cF,\cU}(X)) \iso A^{\cF,\cU}(X) |
|
160 \end{equation*} |
|
161 \end{property} |
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162 |
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163 \begin{property}[Hochschild homology when $X=S^1$] |
|
164 \label{property:hochschild}% |
|
165 The blob complex for a $1$-category $\cC$ on the circle is |
|
166 quasi-isomorphic to the Hochschild complex. |
|
167 \begin{equation*} |
|
168 \xymatrix{\bc_*^{\cC}(S^1) \ar[r]^{\iso}_{\text{qi}} & HC_*(\cC)} |
|
169 \end{equation*} |
|
170 \end{property} |
|
171 |
|
172 \begin{property}[Evaluation map] |
|
173 \label{property:evaluation}% |
|
174 There is an `evaluation' chain map |
|
175 \begin{equation*} |
|
176 \ev_X: \CD{X} \tensor \bc_*(X) \to \bc_*(X). |
|
177 \end{equation*} |
|
178 (Here $\CD{X}$ is the singular chain complex of the space of diffeomorphisms of $X$, fixed on $\bdy X$.) |
|
179 |
|
180 Restricted to $C_0(\Diff(X))$ this is just the action of diffeomorphisms described in Property \ref{property:functoriality}. Further, for |
|
181 any codimension $1$-submanifold $Y \subset X$ dividing $X$ into $X_1 \cup_Y X_2$, the following diagram |
|
182 (using the gluing maps described in Property \ref{property:gluing-map}) commutes. |
|
183 \begin{equation*} |
|
184 \xymatrix{ |
|
185 \CD{X} \otimes \bc_*(X) \ar[r]^{\ev_X} & \bc_*(X) \\ |
|
186 \CD{X_1} \otimes \CD{X_2} \otimes \bc_*(X_1) \otimes \bc_*(X_2) |
|
187 \ar@/_4ex/[r]_{\ev_{X_1} \otimes \ev_{X_2}} \ar[u]^{\gl^{\Diff}_Y \otimes \gl_Y} & |
|
188 \bc_*(X_1) \otimes \bc_*(X_2) \ar[u]_{\gl_Y} |
|
189 } |
|
190 \end{equation*} |
|
191 \end{property} |
|
192 |
|
193 \begin{property}[Gluing formula] |
|
194 \label{property:gluing}% |
|
195 \mbox{}% <-- gets the indenting right |
|
196 \begin{itemize} |
|
197 \item For any $(n-1)$-manifold $Y$, the blob homology of $Y \times I$ is |
|
198 naturally an $A_\infty$ category. % We'll write $\bc_*(Y)$ for $\bc_*(Y \times I)$ below. |
|
199 |
|
200 \item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob homology of $X$ is naturally an |
|
201 $A_\infty$ module for $\bc_*(Y \times I)$. |
|
202 |
|
203 \item For any $n$-manifold $X$, with $Y \cup Y^{\text{op}}$ a codimension |
|
204 $0$-submanifold of its boundary, the blob homology of $X'$, obtained from |
|
205 $X$ by gluing along $Y$, is the $A_\infty$ self-tensor product of |
|
206 $\bc_*(X)$ as an $\bc_*(Y \times I)$-bimodule. |
|
207 \begin{equation*} |
|
208 \bc_*(X') \iso \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y \times I)}} |
|
209 \end{equation*} |
|
210 \todo{How do you write self tensor product?} |
|
211 \end{itemize} |
|
212 \end{property} |
|
213 |
|
214 Properties \ref{property:functoriality}, \ref{property:gluing-map} and \ref{property:skein-modules} will be immediate from the definition given in |
|
215 \S \ref{sec:blob-definition}, and we'll recall them at the appropriate points there. \todo{Make sure this gets done.} |
|
216 Properties \ref{property:disjoint-union} and \ref{property:contractibility} are established in \S \ref{sec:basic-properties}. |
|
217 Property \ref{property:hochschild} is established in \S \ref{sec:hochschild}, Property \ref{property:evaluation} in \S \ref{sec:evaluation}, |
|
218 and Property \ref{property:gluing} in \S \ref{sec:gluing}. |
|
219 |
95 \section{Definitions} |
220 \section{Definitions} |
96 |
221 \label{sec:definitions} |
97 \subsection{Fields} |
222 |
|
223 \subsection{Systems of fields} |
|
224 \label{sec:fields} |
98 |
225 |
99 Fix a top dimension $n$. |
226 Fix a top dimension $n$. |
100 |
227 |
101 A {\it system of fields} |
228 A {\it system of fields} |
102 \nn{maybe should look for better name; but this is the name I use elsewhere} |
229 \nn{maybe should look for better name; but this is the name I use elsewhere} |
581 |
711 |
582 \section{Hochschild homology when $n=1$} |
712 \section{Hochschild homology when $n=1$} |
583 \label{sec:hochschild} |
713 \label{sec:hochschild} |
584 \input{text/hochschild} |
714 \input{text/hochschild} |
585 |
715 |
586 \section{Action of $C_*(\Diff(X))$} \label{diffsect} |
716 \section{Action of $\CD{X}$} |
|
717 \label{sec:evaluation} |
587 |
718 |
588 Let $CD_*(X)$ denote $C_*(\Diff(X))$, the singular chain complex of |
719 Let $CD_*(X)$ denote $C_*(\Diff(X))$, the singular chain complex of |
589 the space of diffeomorphisms |
720 the space of diffeomorphisms |
590 of the $n$-manifold $X$ (fixed on $\bd X$). |
721 of the $n$-manifold $X$ (fixed on $\bd X$). |
591 For convenience, we will permit the singular cells generating $CD_*(X)$ to be more general |
722 For convenience, we will permit the singular cells generating $CD_*(X)$ to be more general |
779 |
910 |
780 \medskip |
911 \medskip |
781 |
912 |
782 \nn{say something about associativity here} |
913 \nn{say something about associativity here} |
783 |
914 |
784 |
915 \section{Gluing} |
785 |
916 \label{sec:gluing}% |
786 |
917 |
787 \section{Families of Diffeomorphisms} \label{fam_diff_sect} |
918 \subsection{`Topological' $A_\infty$ $n$-categories} |
|
919 \label{sec:topological-A-infty}% |
|
920 |
|
921 This section prepares the ground for establishing Property \ref{property:gluing} by defining the notion of a \emph{topological $A_\infty$ |
|
922 $n$-category}. The main result of this section is |
|
923 |
|
924 \begin{thm} |
|
925 Topological $A_\infty$ $1$-categories are equivalent to `standard' |
|
926 $A_\infty$ $1$-categories. |
|
927 \end{thm} |
|
928 |
|
929 |
|
930 |
|
931 \nn{Need to let the input $n$-category $C$ be a graded thing (e.g. DG |
|
932 $n$-category or $A_\infty$ $n$-category). DG $n$-category case is pretty |
|
933 easy, I think, so maybe it should be done earlier??} |
|
934 |
|
935 \bigskip |
|
936 |
|
937 Outline: |
|
938 \begin{itemize} |
|
939 \item recall defs of $A_\infty$ category (1-category only), modules, (self-) tensor product. |
|
940 use graphical/tree point of view, rather than following Keller exactly |
|
941 \item define blob complex in $A_\infty$ case; fat mapping cones? tree decoration? |
|
942 \item topological $A_\infty$ cat def (maybe this should go first); also modules gluing |
|
943 \item motivating example: $C_*(\maps(X, M))$ |
|
944 \item maybe incorporate dual point of view (for $n=1$), where points get |
|
945 object labels and intervals get 1-morphism labels |
|
946 \end{itemize} |
|
947 |
|
948 |
|
949 \subsection{$A_\infty$ action on the boundary} |
|
950 |
|
951 Let $Y$ be an $n{-}1$-manifold. |
|
952 The collection of complexes $\{\bc_*(Y\times I; a, b)\}$, where $a, b \in \cC(Y)$ are boundary |
|
953 conditions on $\bd(Y\times I) = Y\times \{0\} \cup Y\times\{1\}$, has the structure |
|
954 of an $A_\infty$ category. |
|
955 |
|
956 Composition of morphisms (multiplication) depends of a choice of homeomorphism |
|
957 $I\cup I \cong I$. Given this choice, gluing gives a map |
|
958 \eq{ |
|
959 \bc_*(Y\times I; a, b) \otimes \bc_*(Y\times I; b, c) \to \bc_*(Y\times (I\cup I); a, c) |
|
960 \cong \bc_*(Y\times I; a, c) |
|
961 } |
|
962 Using (\ref{CDprop}) and the inclusion $\Diff(I) \sub \Diff(Y\times I)$ gives the various |
|
963 higher associators of the $A_\infty$ structure, more or less canonically. |
|
964 |
|
965 \nn{is this obvious? does more need to be said?} |
|
966 |
|
967 Let $\cA(Y)$ denote the $A_\infty$ category $\bc_*(Y\times I; \cdot, \cdot)$. |
|
968 |
|
969 Similarly, if $Y \sub \bd X$, a choice of collaring homeomorphism |
|
970 $(Y\times I) \cup_Y X \cong X$ gives the collection of complexes $\bc_*(X; r, a)$ |
|
971 (variable $a \in \cC(Y)$; fixed $r \in \cC(\bd X \setmin Y)$) the structure of a representation of the |
|
972 $A_\infty$ category $\{\bc_*(Y\times I; \cdot, \cdot)\}$. |
|
973 Again the higher associators come from the action of $\Diff(I)$ on a collar neighborhood |
|
974 of $Y$ in $X$. |
|
975 |
|
976 In the next section we use the above $A_\infty$ actions to state and prove |
|
977 a gluing theorem for the blob complexes of $n$-manifolds. |
|
978 |
|
979 |
|
980 \subsection{The gluing formula} |
|
981 |
|
982 Let $Y$ be an $n{-}1$-manifold and let $X$ be an $n$-manifold with a copy |
|
983 of $Y \du -Y$ contained in its boundary. |
|
984 Gluing the two copies of $Y$ together we obtain a new $n$-manifold $X\sgl$. |
|
985 We wish to describe the blob complex of $X\sgl$ in terms of the blob complex |
|
986 of $X$. |
|
987 More precisely, we want to describe $\bc_*(X\sgl; c\sgl)$, |
|
988 where $c\sgl \in \cC(\bd X\sgl)$, |
|
989 in terms of the collection $\{\bc_*(X; c, \cdot, \cdot)\}$, thought of as a representation |
|
990 of the $A_\infty$ category $\cA(Y\du-Y) \cong \cA(Y)\times \cA(Y)\op$. |
|
991 |
|
992 \begin{thm} |
|
993 $\bc_*(X\sgl; c\sgl)$ is quasi-isomorphic to the the self tensor product |
|
994 of $\{\bc_*(X; c, \cdot, \cdot)\}$ over $\cA(Y)$. |
|
995 \end{thm} |
|
996 |
|
997 The proof will occupy the remainder of this section. |
|
998 |
|
999 \nn{...} |
|
1000 |
|
1001 \bigskip |
|
1002 |
|
1003 \nn{need to define/recall def of (self) tensor product over an $A_\infty$ category} |
|
1004 |
|
1005 |
|
1006 |
|
1007 |
|
1008 \appendix |
|
1009 |
|
1010 \section{Families of Diffeomorphisms} \label{sec:localising} |
788 |
1011 |
789 |
1012 |
790 Lo, the proof of Lemma (\ref{extension_lemma}): |
1013 Lo, the proof of Lemma (\ref{extension_lemma}): |
791 |
1014 |
792 \nn{should this be an appendix instead?} |
1015 \nn{should this be an appendix instead?} |
970 |
1193 |
971 \nn{this completes proof} |
1194 \nn{this completes proof} |
972 |
1195 |
973 \input{text/explicit.tex} |
1196 \input{text/explicit.tex} |
974 |
1197 |
975 |
1198 % ---------------------------------------------------------------- |
976 \section{$A_\infty$ action on the boundary} |
1199 \newcommand{\urlprefix}{} |
977 |
1200 \bibliographystyle{gtart} |
978 Let $Y$ be an $n{-}1$-manifold. |
1201 %Included for winedt: |
979 The collection of complexes $\{\bc_*(Y\times I; a, b)\}$, where $a, b \in \cC(Y)$ are boundary |
1202 %input "bibliography/bibliography.bib" |
980 conditions on $\bd(Y\times I) = Y\times \{0\} \cup Y\times\{1\}$, has the structure |
1203 \bibliography{bibliography/bibliography} |
981 of an $A_\infty$ category. |
1204 % ---------------------------------------------------------------- |
982 |
1205 |
983 Composition of morphisms (multiplication) depends of a choice of homeomorphism |
1206 This paper is available online at \arxiv{?????}, and at |
984 $I\cup I \cong I$. Given this choice, gluing gives a map |
1207 \url{http://tqft.net/blobs}. |
985 \eq{ |
1208 |
986 \bc_*(Y\times I; a, b) \otimes \bc_*(Y\times I; b, c) \to \bc_*(Y\times (I\cup I); a, c) |
1209 % A GTART necessity: |
987 \cong \bc_*(Y\times I; a, c) |
1210 % \Addresses |
988 } |
1211 % ---------------------------------------------------------------- |
989 Using (\ref{CDprop}) and the inclusion $\Diff(I) \sub \Diff(Y\times I)$ gives the various |
|
990 higher associators of the $A_\infty$ structure, more or less canonically. |
|
991 |
|
992 \nn{is this obvious? does more need to be said?} |
|
993 |
|
994 Let $\cA(Y)$ denote the $A_\infty$ category $\bc_*(Y\times I; \cdot, \cdot)$. |
|
995 |
|
996 Similarly, if $Y \sub \bd X$, a choice of collaring homeomorphism |
|
997 $(Y\times I) \cup_Y X \cong X$ gives the collection of complexes $\bc_*(X; r, a)$ |
|
998 (variable $a \in \cC(Y)$; fixed $r \in \cC(\bd X \setmin Y)$) the structure of a representation of the |
|
999 $A_\infty$ category $\{\bc_*(Y\times I; \cdot, \cdot)\}$. |
|
1000 Again the higher associators come from the action of $\Diff(I)$ on a collar neighborhood |
|
1001 of $Y$ in $X$. |
|
1002 |
|
1003 In the next section we use the above $A_\infty$ actions to state and prove |
|
1004 a gluing theorem for the blob complexes of $n$-manifolds. |
|
1005 |
|
1006 |
|
1007 |
|
1008 |
|
1009 |
|
1010 |
|
1011 |
|
1012 \section{Gluing} \label{gluesect} |
|
1013 |
|
1014 Let $Y$ be an $n{-}1$-manifold and let $X$ be an $n$-manifold with a copy |
|
1015 of $Y \du -Y$ contained in its boundary. |
|
1016 Gluing the two copies of $Y$ together we obtain a new $n$-manifold $X\sgl$. |
|
1017 We wish to describe the blob complex of $X\sgl$ in terms of the blob complex |
|
1018 of $X$. |
|
1019 More precisely, we want to describe $\bc_*(X\sgl; c\sgl)$, |
|
1020 where $c\sgl \in \cC(\bd X\sgl)$, |
|
1021 in terms of the collection $\{\bc_*(X; c, \cdot, \cdot)\}$, thought of as a representation |
|
1022 of the $A_\infty$ category $\cA(Y\du-Y) \cong \cA(Y)\times \cA(Y)\op$. |
|
1023 |
|
1024 \begin{thm} |
|
1025 $\bc_*(X\sgl; c\sgl)$ is quasi-isomorphic to the the self tensor product |
|
1026 of $\{\bc_*(X; c, \cdot, \cdot)\}$ over $\cA(Y)$. |
|
1027 \end{thm} |
|
1028 |
|
1029 The proof will occupy the remainder of this section. |
|
1030 |
|
1031 \nn{...} |
|
1032 |
|
1033 \bigskip |
|
1034 |
|
1035 \nn{need to define/recall def of (self) tensor product over an $A_\infty$ category} |
|
1036 |
|
1037 |
|
1038 |
|
1039 |
|
1040 |
|
1041 \section{Extension to ...} |
|
1042 |
|
1043 \nn{Need to let the input $n$-category $C$ be a graded thing |
|
1044 (e.g. DG $n$-category or $A_\infty$ $n$-category). |
|
1045 DG $n$-category case is pretty easy, I think, so maybe it should be done earlier?? |
|
1046 Also, $A_\infty$ stuff (this section) should go before gluing section.} |
|
1047 |
|
1048 \bigskip |
|
1049 |
|
1050 Outline: |
|
1051 \begin{itemize} |
|
1052 \item recall defs of $A_\infty$ category (1-category only), modules, (self-) tensor product. |
|
1053 use graphical/tree point of view, rather than following Keller exactly |
|
1054 \item define blob complex in $A_\infty$ case; fat mapping cones? tree decoration? |
|
1055 \item topological $A_\infty$ cat def (maybe this should go first); also modules gluing |
|
1056 \item motivating example: $C_*(\maps(X, M))$ |
|
1057 \item maybe incorporate dual point of view (for $n=1$), where points get |
|
1058 object labels and intervals get 1-morphism labels |
|
1059 \end{itemize} |
|
1060 |
|
1061 |
|
1062 |
|
1063 |
|
1064 |
|
1065 |
|
1066 |
|
1067 |
|
1068 |
|
1069 |
|
1070 |
|
1071 \section{What else?...} |
|
1072 |
|
1073 \begin{itemize} |
|
1074 \item Derive Hochschild standard results from blob point of view? |
|
1075 \item $n=2$ examples |
|
1076 \item Kh |
|
1077 \item dimension $n+1$ (generalized Deligne conjecture?) |
|
1078 \item should be clear about PL vs Diff; probably PL is better |
|
1079 (or maybe not) |
|
1080 \item say what we mean by $n$-category, $A_\infty$ or $E_\infty$ $n$-category |
|
1081 \item something about higher derived coend things (derived 2-coend, e.g.) |
|
1082 \end{itemize} |
|
1083 |
|
1084 |
|
1085 |
|
1086 |
|
1087 |
|
1088 \end{document} |
1212 \end{document} |
|
1213 % ---------------------------------------------------------------- |
|
1214 |
1089 |
1215 |
1090 |
1216 |
1091 |
1217 |
1092 %Recall that for $n$-category picture fields there is an evaluation map |
1218 %Recall that for $n$-category picture fields there is an evaluation map |
1093 %$m: \bc_0(B^n; c, c') \to \mor(c, c')$. |
1219 %$m: \bc_0(B^n; c, c') \to \mor(c, c')$. |