61 \end{itemize} |
61 \end{itemize} |
62 \end{itemize} |
62 \end{itemize} |
63 |
63 |
64 } %end \noop |
64 } %end \noop |
65 |
65 |
66 \section{Introduction} |
66 |
67 |
67 |
68 [Outline for intro] |
68 \input{text/intro.tex} |
69 \begin{itemize} |
69 |
70 \item Starting point: TQFTs via fields and local relations. |
|
71 This gives a satisfactory treatment for semisimple TQFTs |
|
72 (i.e.\ TQFTs for which the cylinder 1-category associated to an |
|
73 $n{-}1$-manifold $Y$ is semisimple for all $Y$). |
|
74 \item For non-semiemple TQFTs, this approach is less satisfactory. |
|
75 Our main motivating example (though we will not develop it in this paper) |
|
76 is the $4{+}1$-dimensional TQFT associated to Khovanov homology. |
|
77 It associates a bigraded vector space $A_{Kh}(W^4, L)$ to a 4-manifold $W$ together |
|
78 with a link $L \subset \bd W$. |
|
79 The original Khovanov homology of a link in $S^3$ is recovered as $A_{Kh}(B^4, L)$. |
|
80 \item How would we go about computing $A_{Kh}(W^4, L)$? |
|
81 For $A_{Kh}(B^4, L)$, the main tool is the exact triangle (long exact sequence) |
|
82 \nn{... $L_1, L_2, L_3$}. |
|
83 Unfortunately, the exactness breaks if we glue $B^4$ to itself and attempt |
|
84 to compute $A_{Kh}(S^1\times B^3, L)$. |
|
85 According to the gluing theorem for TQFTs-via-fields, gluing along $B^3 \subset \bd B^4$ |
|
86 corresponds to taking a coend (self tensor product) over the cylinder category |
|
87 associated to $B^3$ (with appropriate boundary conditions). |
|
88 The coend is not an exact functor, so the exactness of the triangle breaks. |
|
89 \item The obvious solution to this problem is to replace the coend with its derived counterpart. |
|
90 This presumably works fine for $S^1\times B^3$ (the answer being the Hochschild homology |
|
91 of an appropriate bimodule), but for more complicated 4-manifolds this leaves much to be desired. |
|
92 If we build our manifold up via a handle decomposition, the computation |
|
93 would be a sequence of derived coends. |
|
94 A different handle decomposition of the same manifold would yield a different |
|
95 sequence of derived coends. |
|
96 To show that our definition in terms of derived coends is well-defined, we |
|
97 would need to show that the above two sequences of derived coends yield the same answer. |
|
98 This is probably not easy to do. |
|
99 \item Instead, we would prefer a definition for a derived version of $A_{Kh}(W^4, L)$ |
|
100 which is manifestly invariant. |
|
101 (That is, a definition that does not |
|
102 involve choosing a decomposition of $W$. |
|
103 After all, one of the virtues of our starting point --- TQFTs via field and local relations --- |
|
104 is that it has just this sort of manifest invariance.) |
|
105 \item The solution is to replace $A_{Kh}(W^4, L)$, which is a quotient |
|
106 \[ |
|
107 \text{linear combinations of fields} \;\big/\; \text{local relations} , |
|
108 \] |
|
109 with an appropriately free resolution (the ``blob complex") |
|
110 \[ |
|
111 \cdots\to \bc_2(W, L) \to \bc_1(W, L) \to \bc_0(W, L) . |
|
112 \] |
|
113 Here $\bc_0$ is linear combinations of fields on $W$, |
|
114 $\bc_1$ is linear combinations of local relations on $W$, |
|
115 $\bc_2$ is linear combinations of relations amongst relations on $W$, |
|
116 and so on. |
|
117 \item None of the above ideas depend on the details of the Khovanov homology example, |
|
118 so we develop the general theory in the paper and postpone specific applications |
|
119 to later papers. |
|
120 \item The blob complex enjoys the following nice properties \nn{...} |
|
121 \end{itemize} |
|
122 |
|
123 \bigskip |
|
124 \hrule |
|
125 \bigskip |
|
126 |
|
127 We then show that blob homology enjoys the following |
|
128 \ref{property:gluing} properties. |
|
129 |
|
130 \begin{property}[Functoriality] |
|
131 \label{property:functoriality}% |
|
132 Blob homology is functorial with respect to diffeomorphisms. That is, fixing an $n$-dimensional system of fields $\cF$ and local relations $\cU$, the association |
|
133 \begin{equation*} |
|
134 X \mapsto \bc_*^{\cF,\cU}(X) |
|
135 \end{equation*} |
|
136 is a functor from $n$-manifolds and diffeomorphisms between them to chain complexes and isomorphisms between them. |
|
137 \end{property} |
|
138 |
|
139 \begin{property}[Disjoint union] |
|
140 \label{property:disjoint-union} |
|
141 The blob complex of a disjoint union is naturally the tensor product of the blob complexes. |
|
142 \begin{equation*} |
|
143 \bc_*(X_1 \du X_2) \iso \bc_*(X_1) \tensor \bc_*(X_2) |
|
144 \end{equation*} |
|
145 \end{property} |
|
146 |
|
147 \begin{property}[A map for gluing] |
|
148 \label{property:gluing-map}% |
|
149 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 $\bdy X_2$, |
|
150 there is a chain map |
|
151 \begin{equation*} |
|
152 \gl_Y: \bc_*(X_1) \tensor \bc_*(X_2) \to \bc_*(X_1 \cup_Y X_2). |
|
153 \end{equation*} |
|
154 \end{property} |
|
155 |
|
156 \begin{property}[Contractibility] |
|
157 \label{property:contractibility}% |
|
158 \todo{Err, requires a splitting?} |
|
159 The blob complex for an $n$-category on an $n$-ball is quasi-isomorphic to its $0$-th homology. |
|
160 \begin{equation} |
|
161 \xymatrix{\bc_*^{\cC}(B^n) \ar[r]^{\iso}_{\text{qi}} & H_0(\bc_*^{\cC}(B^n))} |
|
162 \end{equation} |
|
163 \todo{Say that this is just the original $n$-category?} |
|
164 \end{property} |
|
165 |
|
166 \begin{property}[Skein modules] |
|
167 \label{property:skein-modules}% |
|
168 The $0$-th blob homology of $X$ is the usual |
|
169 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
|
170 by $(\cF,\cU)$. (See \S \ref{sec:local-relations}.) |
|
171 \begin{equation*} |
|
172 H_0(\bc_*^{\cF,\cU}(X)) \iso A^{\cF,\cU}(X) |
|
173 \end{equation*} |
|
174 \end{property} |
|
175 |
|
176 \begin{property}[Hochschild homology when $X=S^1$] |
|
177 \label{property:hochschild}% |
|
178 The blob complex for a $1$-category $\cC$ on the circle is |
|
179 quasi-isomorphic to the Hochschild complex. |
|
180 \begin{equation*} |
|
181 \xymatrix{\bc_*^{\cC}(S^1) \ar[r]^{\iso}_{\text{qi}} & HC_*(\cC)} |
|
182 \end{equation*} |
|
183 \end{property} |
|
184 |
|
185 \begin{property}[Evaluation map] |
|
186 \label{property:evaluation}% |
|
187 There is an `evaluation' chain map |
|
188 \begin{equation*} |
|
189 \ev_X: \CD{X} \tensor \bc_*(X) \to \bc_*(X). |
|
190 \end{equation*} |
|
191 (Here $\CD{X}$ is the singular chain complex of the space of diffeomorphisms of $X$, fixed on $\bdy X$.) |
|
192 |
|
193 Restricted to $C_0(\Diff(X))$ this is just the action of diffeomorphisms described in Property \ref{property:functoriality}. Further, for |
|
194 any codimension $1$-submanifold $Y \subset X$ dividing $X$ into $X_1 \cup_Y X_2$, the following diagram |
|
195 (using the gluing maps described in Property \ref{property:gluing-map}) commutes. |
|
196 \begin{equation*} |
|
197 \xymatrix{ |
|
198 \CD{X} \otimes \bc_*(X) \ar[r]^{\ev_X} & \bc_*(X) \\ |
|
199 \CD{X_1} \otimes \CD{X_2} \otimes \bc_*(X_1) \otimes \bc_*(X_2) |
|
200 \ar@/_4ex/[r]_{\ev_{X_1} \otimes \ev_{X_2}} \ar[u]^{\gl^{\Diff}_Y \otimes \gl_Y} & |
|
201 \bc_*(X_1) \otimes \bc_*(X_2) \ar[u]_{\gl_Y} |
|
202 } |
|
203 \end{equation*} |
|
204 \nn{should probably say something about associativity here (or not?)} |
|
205 \end{property} |
|
206 |
|
207 |
|
208 \begin{property}[Gluing formula] |
|
209 \label{property:gluing}% |
|
210 \mbox{}% <-- gets the indenting right |
|
211 \begin{itemize} |
|
212 \item For any $(n-1)$-manifold $Y$, the blob homology of $Y \times I$ is |
|
213 naturally an $A_\infty$ category. % We'll write $\bc_*(Y)$ for $\bc_*(Y \times I)$ below. |
|
214 |
|
215 \item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob homology of $X$ is naturally an |
|
216 $A_\infty$ module for $\bc_*(Y \times I)$. |
|
217 |
|
218 \item For any $n$-manifold $X$, with $Y \cup Y^{\text{op}}$ a codimension |
|
219 $0$-submanifold of its boundary, the blob homology of $X'$, obtained from |
|
220 $X$ by gluing along $Y$, is the $A_\infty$ self-tensor product of |
|
221 $\bc_*(X)$ as an $\bc_*(Y \times I)$-bimodule. |
|
222 \begin{equation*} |
|
223 \bc_*(X') \iso \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y \times I)}} \!\!\!\!\!\!\xymatrix{ \ar@(ru,rd)@<-1ex>[]} |
|
224 \end{equation*} |
|
225 \end{itemize} |
|
226 \end{property} |
|
227 |
|
228 \nn{add product formula? $n$-dimensional fat graph operad stuff?} |
|
229 |
|
230 Properties \ref{property:functoriality}, \ref{property:gluing-map} and \ref{property:skein-modules} will be immediate from the definition given in |
|
231 \S \ref{sec:blob-definition}, and we'll recall them at the appropriate points there. \todo{Make sure this gets done.} |
|
232 Properties \ref{property:disjoint-union} and \ref{property:contractibility} are established in \S \ref{sec:basic-properties}. |
|
233 Property \ref{property:hochschild} is established in \S \ref{sec:hochschild}, Property \ref{property:evaluation} in \S \ref{sec:evaluation}, |
|
234 and Property \ref{property:gluing} in \S \ref{sec:gluing}. |
|
235 |
70 |
236 \section{Definitions} |
71 \section{Definitions} |
237 \label{sec:definitions} |
72 \label{sec:definitions} |
238 |
73 |
239 \subsection{Systems of fields} |
74 \subsection{Systems of fields} |
1017 |
852 |
1018 |
853 |
1019 |
854 |
1020 \appendix |
855 \appendix |
1021 |
856 |
1022 \section{Families of Diffeomorphisms} \label{sec:localising} |
857 \input{text/famodiff.tex} |
1023 |
|
1024 |
|
1025 Lo, the proof of Lemma (\ref{extension_lemma}): |
|
1026 |
|
1027 \nn{should this be an appendix instead?} |
|
1028 |
|
1029 \nn{for pedagogical reasons, should do $k=1,2$ cases first; probably do this in |
|
1030 later draft} |
|
1031 |
|
1032 \nn{not sure what the best way to deal with boundary is; for now just give main argument, worry |
|
1033 about boundary later} |
|
1034 |
|
1035 Recall that we are given |
|
1036 an open cover $\cU = \{U_\alpha\}$ and an |
|
1037 $x \in CD_k(X)$ such that $\bd x$ is adapted to $\cU$. |
|
1038 We must find a homotopy of $x$ (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$. |
|
1039 |
|
1040 Let $\{r_\alpha : X \to [0,1]\}$ be a partition of unity for $\cU$. |
|
1041 |
|
1042 As a first approximation to the argument we will eventually make, let's replace $x$ |
|
1043 with a single singular cell |
|
1044 \eq{ |
|
1045 f: P \times X \to X . |
|
1046 } |
|
1047 Also, we'll ignore for now issues around $\bd P$. |
|
1048 |
|
1049 Our homotopy will have the form |
|
1050 \eqar{ |
|
1051 F: I \times P \times X &\to& X \\ |
|
1052 (t, p, x) &\mapsto& f(u(t, p, x), x) |
|
1053 } |
|
1054 for some function |
|
1055 \eq{ |
|
1056 u : I \times P \times X \to P . |
|
1057 } |
|
1058 First we describe $u$, then we argue that it does what we want it to do. |
|
1059 |
|
1060 For each cover index $\alpha$ choose a cell decomposition $K_\alpha$ of $P$. |
|
1061 The various $K_\alpha$ should be in general position with respect to each other. |
|
1062 We will see below that the $K_\alpha$'s need to be sufficiently fine in order |
|
1063 to insure that $F$ above is a homotopy through diffeomorphisms of $X$ and not |
|
1064 merely a homotopy through maps $X\to X$. |
|
1065 |
|
1066 Let $L$ be the union of all the $K_\alpha$'s. |
|
1067 $L$ is itself a cell decomposition of $P$. |
|
1068 \nn{next two sentences not needed?} |
|
1069 To each cell $a$ of $L$ we associate the tuple $(c_\alpha)$, |
|
1070 where $c_\alpha$ is the codimension of the cell of $K_\alpha$ which contains $c$. |
|
1071 Since the $K_\alpha$'s are in general position, we have $\sum c_\alpha \le k$. |
|
1072 |
|
1073 Let $J$ denote the handle decomposition of $P$ corresponding to $L$. |
|
1074 Each $i$-handle $C$ of $J$ has an $i$-dimensional tangential coordinate and, |
|
1075 more importantly, a $k{-}i$-dimensional normal coordinate. |
|
1076 |
|
1077 For each (top-dimensional) $k$-cell $c$ of each $K_\alpha$, choose a point $p_c \in c \sub P$. |
|
1078 Let $D$ be a $k$-handle of $J$, and let $D$ also denote the corresponding |
|
1079 $k$-cell of $L$. |
|
1080 To $D$ we associate the tuple $(c_\alpha)$ of $k$-cells of the $K_\alpha$'s |
|
1081 which contain $d$, and also the corresponding tuple $(p_{c_\alpha})$ of points in $P$. |
|
1082 |
|
1083 For $p \in D$ we define |
|
1084 \eq{ |
|
1085 u(t, p, x) = (1-t)p + t \sum_\alpha r_\alpha(x) p_{c_\alpha} . |
|
1086 } |
|
1087 (Recall that $P$ is a single linear cell, so the weighted average of points of $P$ |
|
1088 makes sense.) |
|
1089 |
|
1090 So far we have defined $u(t, p, x)$ when $p$ lies in a $k$-handle of $J$. |
|
1091 For handles of $J$ of index less than $k$, we will define $u$ to |
|
1092 interpolate between the values on $k$-handles defined above. |
|
1093 |
|
1094 If $p$ lies in a $k{-}1$-handle $E$, let $\eta : E \to [0,1]$ be the normal coordinate |
|
1095 of $E$. |
|
1096 In particular, $\eta$ is equal to 0 or 1 only at the intersection of $E$ |
|
1097 with a $k$-handle. |
|
1098 Let $\beta$ be the index of the $K_\beta$ containing the $k{-}1$-cell |
|
1099 corresponding to $E$. |
|
1100 Let $q_0, q_1 \in P$ be the points associated to the two $k$-cells of $K_\beta$ |
|
1101 adjacent to the $k{-}1$-cell corresponding to $E$. |
|
1102 For $p \in E$, define |
|
1103 \eq{ |
|
1104 u(t, p, x) = (1-t)p + t \left( \sum_{\alpha \ne \beta} r_\alpha(x) p_{c_\alpha} |
|
1105 + r_\beta(x) (\eta(p) q_1 + (1-\eta(p)) q_0) \right) . |
|
1106 } |
|
1107 |
|
1108 In general, for $E$ a $k{-}j$-handle, there is a normal coordinate |
|
1109 $\eta: E \to R$, where $R$ is some $j$-dimensional polyhedron. |
|
1110 The vertices of $R$ are associated to $k$-cells of the $K_\alpha$, and thence to points of $P$. |
|
1111 If we triangulate $R$ (without introducing new vertices), we can linearly extend |
|
1112 a map from the vertices of $R$ into $P$ to a map of all of $R$ into $P$. |
|
1113 Let $\cN$ be the set of all $\beta$ for which $K_\beta$ has a $k$-cell whose boundary meets |
|
1114 the $k{-}j$-cell corresponding to $E$. |
|
1115 For each $\beta \in \cN$, let $\{q_{\beta i}\}$ be the set of points in $P$ associated to the aforementioned $k$-cells. |
|
1116 Now define, for $p \in E$, |
|
1117 \eq{ |
|
1118 u(t, p, x) = (1-t)p + t \left( |
|
1119 \sum_{\alpha \notin \cN} r_\alpha(x) p_{c_\alpha} |
|
1120 + \sum_{\beta \in \cN} r_\beta(x) \left( \sum_i \eta_{\beta i}(p) \cdot q_{\beta i} \right) |
|
1121 \right) . |
|
1122 } |
|
1123 Here $\eta_{\beta i}(p)$ is the weight given to $q_{\beta i}$ by the linear extension |
|
1124 mentioned above. |
|
1125 |
|
1126 This completes the definition of $u: I \times P \times X \to P$. |
|
1127 |
|
1128 \medskip |
|
1129 |
|
1130 Next we verify that $u$ has the desired properties. |
|
1131 |
|
1132 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 $x$. |
|
1133 Therefore $F$ is a homotopy from $f$ to something. |
|
1134 |
|
1135 Next we show that if the $K_\alpha$'s are sufficiently fine cell decompositions, |
|
1136 then $F$ is a homotopy through diffeomorphisms. |
|
1137 We must show that the derivative $\pd{F}{x}(t, p, x)$ is non-singular for all $(t, p, x)$. |
|
1138 We have |
|
1139 \eq{ |
|
1140 % \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) . |
|
1141 \pd{F}{x} = \pd{f}{x} + \pd{f}{p} \pd{u}{x} . |
|
1142 } |
|
1143 Since $f$ is a family of diffeomorphisms, $\pd{f}{x}$ is non-singular and |
|
1144 \nn{bounded away from zero, or something like that}. |
|
1145 (Recall that $X$ and $P$ are compact.) |
|
1146 Also, $\pd{f}{p}$ is bounded. |
|
1147 So if we can insure that $\pd{u}{x}$ is sufficiently small, we are done. |
|
1148 It follows from Equation xxxx above that $\pd{u}{x}$ depends on $\pd{r_\alpha}{x}$ |
|
1149 (which is bounded) |
|
1150 and the differences amongst the various $p_{c_\alpha}$'s and $q_{\beta i}$'s. |
|
1151 These differences are small if the cell decompositions $K_\alpha$ are sufficiently fine. |
|
1152 This completes the proof that $F$ is a homotopy through diffeomorphisms. |
|
1153 |
|
1154 \medskip |
|
1155 |
|
1156 Next we show that for each handle $D \sub P$, $F(1, \cdot, \cdot) : D\times X \to X$ |
|
1157 is a singular cell adapted to $\cU$. |
|
1158 This will complete the proof of the lemma. |
|
1159 \nn{except for boundary issues and the `$P$ is a cell' assumption} |
|
1160 |
|
1161 Let $j$ be the codimension of $D$. |
|
1162 (Or rather, the codimension of its corresponding cell. From now on we will not make a distinction |
|
1163 between handle and corresponding cell.) |
|
1164 Then $j = j_1 + \cdots + j_m$, $0 \le m \le k$, |
|
1165 where the $j_i$'s are the codimensions of the $K_\alpha$ |
|
1166 cells of codimension greater than 0 which intersect to form $D$. |
|
1167 We will show that |
|
1168 if the relevant $U_\alpha$'s are disjoint, then |
|
1169 $F(1, \cdot, \cdot) : D\times X \to X$ |
|
1170 is a product of singular cells of dimensions $j_1, \ldots, j_m$. |
|
1171 If some of the relevant $U_\alpha$'s intersect, then we will get a product of singular |
|
1172 cells whose dimensions correspond to a partition of the $j_i$'s. |
|
1173 We will consider some simple special cases first, then do the general case. |
|
1174 |
|
1175 First consider the case $j=0$ (and $m=0$). |
|
1176 A quick look at Equation xxxx above shows that $u(1, p, x)$, and hence $F(1, p, x)$, |
|
1177 is independent of $p \in P$. |
|
1178 So the corresponding map $D \to \Diff(X)$ is constant. |
|
1179 |
|
1180 Next consider the case $j = 1$ (and $m=1$, $j_1=1$). |
|
1181 Now Equation yyyy applies. |
|
1182 We can write $D = D'\times I$, where the normal coordinate $\eta$ is constant on $D'$. |
|
1183 It follows that the singular cell $D \to \Diff(X)$ can be written as a product |
|
1184 of a constant map $D' \to \Diff(X)$ and a singular 1-cell $I \to \Diff(X)$. |
|
1185 The singular 1-cell is supported on $U_\beta$, since $r_\beta = 0$ outside of this set. |
|
1186 |
|
1187 Next case: $j=2$, $m=1$, $j_1 = 2$. |
|
1188 This is similar to the previous case, except that the normal bundle is 2-dimensional instead of |
|
1189 1-dimensional. |
|
1190 We have that $D \to \Diff(X)$ is a product of a constant singular $k{-}2$-cell |
|
1191 and a 2-cell with support $U_\beta$. |
|
1192 |
|
1193 Next case: $j=2$, $m=2$, $j_1 = j_2 = 1$. |
|
1194 In this case the codimension 2 cell $D$ is the intersection of two |
|
1195 codimension 1 cells, from $K_\beta$ and $K_\gamma$. |
|
1196 We can write $D = D' \times I \times I$, where the normal coordinates are constant |
|
1197 on $D'$, and the two $I$ factors correspond to $\beta$ and $\gamma$. |
|
1198 If $U_\beta$ and $U_\gamma$ are disjoint, then we can factor $D$ into a constant $k{-}2$-cell and |
|
1199 two 1-cells, supported on $U_\beta$ and $U_\gamma$ respectively. |
|
1200 If $U_\beta$ and $U_\gamma$ intersect, then we can factor $D$ into a constant $k{-}2$-cell and |
|
1201 a 2-cell supported on $U_\beta \cup U_\gamma$. |
|
1202 \nn{need to check that this is true} |
|
1203 |
|
1204 \nn{finally, general case...} |
|
1205 |
|
1206 \nn{this completes proof} |
|
1207 |
|
1208 \input{text/explicit.tex} |
|
1209 |
858 |
1210 \section{Comparing definitions of $A_\infty$ algebras} |
859 \section{Comparing definitions of $A_\infty$ algebras} |
1211 \label{sec:comparing-A-infty} |
860 \label{sec:comparing-A-infty} |
1212 In this section, we make contact between the usual definition of an $A_\infty$ algebra and our definition of a topological $A_\infty$ algebra, from Definition \ref{defn:topological-Ainfty-category}. |
861 In this section, we make contact between the usual definition of an $A_\infty$ algebra and our definition of a topological $A_\infty$ algebra, from Definition \ref{defn:topological-Ainfty-category}. |
1213 |
862 |