27 [11 June 2009]

    27 [later than 11 June 2009]

    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}.

  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}