2 |
2 |
3 \section{The blob complex} |
3 \section{The blob complex} |
4 \label{sec:blob-definition} |
4 \label{sec:blob-definition} |
5 |
5 |
6 Let $X$ be an $n$-manifold. |
6 Let $X$ be an $n$-manifold. |
7 Let $\cC$ be a fixed system of fields and local relations. |
7 Let $(\cF,U)$ be a fixed system of fields and local relations. |
8 We'll assume it is enriched over \textbf{Vect}; |
8 We'll assume it is enriched over \textbf{Vect}; |
9 if it is not we can make it so by allowing finite |
9 if it is not we can make it so by allowing finite |
10 linear combinations of elements of $\cC(X; c)$, for fixed $c\in \cC(\bd X)$. |
10 linear combinations of elements of $\cF(X; c)$, for fixed $c\in \cF(\bd X)$. |
11 |
11 |
12 %In this section we will usually suppress boundary conditions on $X$ from the notation, e.g. by writing $\lf(X)$ instead of $\lf(X; c)$. |
12 %In this section we will usually suppress boundary conditions on $X$ from the notation, e.g. by writing $\cF(X)$ instead of $\cF(X; c)$. |
13 |
13 |
14 We want to replace the quotient |
14 We want to replace the quotient |
15 \[ |
15 \[ |
16 A(X) \deq \lf(X) / U(X) |
16 A(X) \deq \cF(X) / U(X) |
17 \] |
17 \] |
18 of Definition \ref{defn:TQFT-invariant} with a resolution |
18 of Definition \ref{defn:TQFT-invariant} with a resolution |
19 \[ |
19 \[ |
20 \cdots \to \bc_2(X) \to \bc_1(X) \to \bc_0(X) . |
20 \cdots \to \bc_2(X) \to \bc_1(X) \to \bc_0(X) . |
21 \] |
21 \] |
30 ``nested'' and ``disjoint'', that are not quite the intuitive ones. |
30 ``nested'' and ``disjoint'', that are not quite the intuitive ones. |
31 Moreover some of the pieces |
31 Moreover some of the pieces |
32 into which we cut manifolds below are not themselves manifolds, and it requires special attention |
32 into which we cut manifolds below are not themselves manifolds, and it requires special attention |
33 to define fields on these pieces. |
33 to define fields on these pieces. |
34 |
34 |
35 We of course define $\bc_0(X) = \lf(X)$. |
35 We of course define $\bc_0(X) = \cF(X)$. |
36 (If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \lf(X; c)$ for each $c \in \lf(\bdy X)$. |
36 (If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \cF(X; c)$ for each $c \in \cF(\bdy X)$. |
37 We'll omit such boundary conditions from the notation in the rest of this section.) |
37 We'll omit such boundary conditions from the notation in the rest of this section.) |
38 In other words, $\bc_0(X)$ is just the vector space of all fields on $X$. |
38 In other words, $\bc_0(X)$ is just the vector space of all fields on $X$. |
39 |
39 |
40 We want the vector space $\bc_1(X)$ to capture |
40 We want the vector space $\bc_1(X)$ to capture |
41 ``the space of all local relations that can be imposed on $\bc_0(X)$". |
41 ``the space of all local relations that can be imposed on $\bc_0(X)$". |
42 Thus we say a $1$-blob diagram consists of: |
42 Thus we say a $1$-blob diagram consists of: |
43 \begin{itemize} |
43 \begin{itemize} |
44 \item An closed ball in $X$ (``blob") $B \sub X$. |
44 \item An closed ball in $X$ (``blob") $B \sub X$. |
45 \item A boundary condition $c \in \cC(\bdy B) = \cC(\bd(X \setmin B))$. |
45 \item A boundary condition $c \in \cF(\bdy B) = \cF(\bd(X \setmin B))$. |
46 \item A field $r \in \cC(X \setmin B; c)$. |
46 \item A field $r \in \cF(X \setmin B; c)$. |
47 \item A local relation field $u \in U(B; c)$. |
47 \item A local relation field $u \in U(B; c)$. |
48 \end{itemize} |
48 \end{itemize} |
49 (See Figure \ref{blob1diagram}.) Since $c$ is implicitly determined by $u$ or $r$, we usually omit it from the notation. |
49 (See Figure \ref{blob1diagram}.) Since $c$ is implicitly determined by $u$ or $r$, we usually omit it from the notation. |
50 \begin{figure}[t]\begin{equation*} |
50 \begin{figure}[t]\begin{equation*} |
51 \mathfig{.6}{definition/single-blob} |
51 \mathfig{.6}{definition/single-blob} |
52 \end{equation*}\caption{A 1-blob diagram.}\label{blob1diagram}\end{figure} |
52 \end{equation*}\caption{A 1-blob diagram.}\label{blob1diagram}\end{figure} |
53 In order to get the linear structure correct, we define |
53 In order to get the linear structure correct, we define |
54 \[ |
54 \[ |
55 \bc_1(X) \deq \bigoplus_B \bigoplus_c U(B; c) \otimes \lf(X \setmin B; c) . |
55 \bc_1(X) \deq \bigoplus_B \bigoplus_c U(B; c) \otimes \cF(X \setmin B; c) . |
56 \] |
56 \] |
57 The first direct sum is indexed by all blobs $B\subset X$, and the second |
57 The first direct sum is indexed by all blobs $B\subset X$, and the second |
58 by all boundary conditions $c \in \cC(\bd B)$. |
58 by all boundary conditions $c \in \cF(\bd B)$. |
59 Note that $\bc_1(X)$ is spanned by 1-blob diagrams $(B, u, r)$. |
59 Note that $\bc_1(X)$ is spanned by 1-blob diagrams $(B, u, r)$. |
60 |
60 |
61 Define the boundary map $\bd : \bc_1(X) \to \bc_0(X)$ by |
61 Define the boundary map $\bd : \bc_1(X) \to \bc_0(X)$ by |
62 \[ |
62 \[ |
63 (B, u, r) \mapsto u\bullet r, |
63 (B, u, r) \mapsto u\bullet r, |
81 local relations encoded in $\bc_1(X)$''. |
81 local relations encoded in $\bc_1(X)$''. |
82 A $2$-blob diagram, comes in one of two types, disjoint and nested. |
82 A $2$-blob diagram, comes in one of two types, disjoint and nested. |
83 A disjoint 2-blob diagram consists of |
83 A disjoint 2-blob diagram consists of |
84 \begin{itemize} |
84 \begin{itemize} |
85 \item A pair of closed balls (blobs) $B_1, B_2 \sub X$ with disjoint interiors. |
85 \item A pair of closed balls (blobs) $B_1, B_2 \sub X$ with disjoint interiors. |
86 \item A field $r \in \cC(X \setmin (B_1 \cup B_2); c_1, c_2)$ |
86 \item A field $r \in \cF(X \setmin (B_1 \cup B_2); c_1, c_2)$ |
87 (where $c_i \in \cC(\bd B_i)$). |
87 (where $c_i \in \cF(\bd B_i)$). |
88 \item Local relation fields $u_i \in U(B_i; c_i)$, $i=1,2$. |
88 \item Local relation fields $u_i \in U(B_i; c_i)$, $i=1,2$. |
89 \end{itemize} |
89 \end{itemize} |
90 (See Figure \ref{blob2ddiagram}.) |
90 (See Figure \ref{blob2ddiagram}.) |
91 \begin{figure}[t]\begin{equation*} |
91 \begin{figure}[t]\begin{equation*} |
92 \mathfig{.6}{definition/disjoint-blobs} |
92 \mathfig{.6}{definition/disjoint-blobs} |
101 It's easy to check that $\bd^2 = 0$. |
101 It's easy to check that $\bd^2 = 0$. |
102 |
102 |
103 A nested 2-blob diagram consists of |
103 A nested 2-blob diagram consists of |
104 \begin{itemize} |
104 \begin{itemize} |
105 \item A pair of nested balls (blobs) $B_1 \subseteq B_2 \subseteq X$. |
105 \item A pair of nested balls (blobs) $B_1 \subseteq B_2 \subseteq X$. |
106 \item A field $r' \in \cC(B_2 \setminus B_1; c_1, c_2)$ |
106 \item A field $r' \in \cF(B_2 \setminus B_1; c_1, c_2)$ |
107 (for some $c_1 \in \cC(\bdy B_1)$ and $c_2 \in \cC(\bdy B_2)$). |
107 (for some $c_1 \in \cF(\bdy B_1)$ and $c_2 \in \cF(\bdy B_2)$). |
108 \item A field $r \in \cC(X \setminus B_2; c_2)$. |
108 \item A field $r \in \cF(X \setminus B_2; c_2)$. |
109 \item A local relation field $u \in U(B_1; c_1)$. |
109 \item A local relation field $u \in U(B_1; c_1)$. |
110 \end{itemize} |
110 \end{itemize} |
111 (See Figure \ref{blob2ndiagram}.) |
111 (See Figure \ref{blob2ndiagram}.) |
112 \begin{figure}[t]\begin{equation*} |
112 \begin{figure}[t]\begin{equation*} |
113 \mathfig{.6}{definition/nested-blobs} |
113 \mathfig{.6}{definition/nested-blobs} |
122 As with the $1$-blob diagrams, in order to get the linear structure correct the actual definition is |
122 As with the $1$-blob diagrams, in order to get the linear structure correct the actual definition is |
123 \begin{eqnarray*} |
123 \begin{eqnarray*} |
124 \bc_2(X) & \deq & |
124 \bc_2(X) & \deq & |
125 \left( |
125 \left( |
126 \bigoplus_{B_1, B_2\; \text{disjoint}} \bigoplus_{c_1, c_2} |
126 \bigoplus_{B_1, B_2\; \text{disjoint}} \bigoplus_{c_1, c_2} |
127 U(B_1; c_1) \otimes U(B_2; c_2) \otimes \lf(X\setmin (B_1\cup B_2); c_1, c_2) |
127 U(B_1; c_1) \otimes U(B_2; c_2) \otimes \cF(X\setmin (B_1\cup B_2); c_1, c_2) |
128 \right) \bigoplus \\ |
128 \right) \bigoplus \\ |
129 && \quad\quad \left( |
129 && \quad\quad \left( |
130 \bigoplus_{B_1 \subset B_2} \bigoplus_{c_1, c_2} |
130 \bigoplus_{B_1 \subset B_2} \bigoplus_{c_1, c_2} |
131 U(B_1; c_1) \otimes \lf(B_2 \setmin B_1; c_1, c_2) \tensor \cC(X \setminus B_2; c_2) |
131 U(B_1; c_1) \otimes \cF(B_2 \setmin B_1; c_1, c_2) \tensor \cF(X \setminus B_2; c_2) |
132 \right) . |
132 \right) . |
133 \end{eqnarray*} |
133 \end{eqnarray*} |
134 % __ (already said this above) |
134 % __ (already said this above) |
135 %For the disjoint blobs, reversing the ordering of $B_1$ and $B_2$ introduces a minus sign |
135 %For the disjoint blobs, reversing the ordering of $B_1$ and $B_2$ introduces a minus sign |
136 %(rather than a new, linearly independent, 2-blob diagram). |
136 %(rather than a new, linearly independent, 2-blob diagram). |
195 \begin{defn} |
195 \begin{defn} |
196 \label{defn:blob-diagram} |
196 \label{defn:blob-diagram} |
197 A $k$-blob diagram on $X$ consists of |
197 A $k$-blob diagram on $X$ consists of |
198 \begin{itemize} |
198 \begin{itemize} |
199 \item a configuration $\{B_1, \ldots B_k\}$ of $k$ blobs in $X$, |
199 \item a configuration $\{B_1, \ldots B_k\}$ of $k$ blobs in $X$, |
200 \item and a field $r \in \cC(X)$ which is splittable along some gluing decomposition compatible with that configuration, |
200 \item and a field $r \in \cF(X)$ which is splittable along some gluing decomposition compatible with that configuration, |
201 \end{itemize} |
201 \end{itemize} |
202 such that |
202 such that |
203 the restriction $u_i$ of $r$ to each twig blob $B_i$ lies in the subspace $U(B_i) \subset \cC(B_i)$. (See Figure \ref{blobkdiagram}.) More precisely, each twig blob $B_i$ is the image of some ball $M_r'$ as above, and it is really the restriction to $M_r'$ that must lie in the subspace $U(M_r')$. |
203 the restriction $u_i$ of $r$ to each twig blob $B_i$ lies in the subspace $U(B_i) \subset \cF(B_i)$. (See Figure \ref{blobkdiagram}.) More precisely, each twig blob $B_i$ is the image of some ball $M_r'$ as above, and it is really the restriction to $M_r'$ that must lie in the subspace $U(M_r')$. |
204 \end{defn} |
204 \end{defn} |
205 \begin{figure}[t]\begin{equation*} |
205 \begin{figure}[t]\begin{equation*} |
206 \mathfig{.7}{definition/k-blobs} |
206 \mathfig{.7}{definition/k-blobs} |
207 \end{equation*}\caption{A $k$-blob diagram.}\label{blobkdiagram}\end{figure} |
207 \end{equation*}\caption{A $k$-blob diagram.}\label{blobkdiagram}\end{figure} |
208 and |
208 and |
239 encloses all the others (equivalently, add a new edge to the root, with the new vertex becoming the root). |
239 encloses all the others (equivalently, add a new edge to the root, with the new vertex becoming the root). |
240 \end{itemize} |
240 \end{itemize} |
241 For example, a diagram of $k$ strictly nested blobs corresponds to a $k$-simplex, while |
241 For example, a diagram of $k$ strictly nested blobs corresponds to a $k$-simplex, while |
242 a diagram of $k$ disjoint blobs corresponds to a $k$-cube. |
242 a diagram of $k$ disjoint blobs corresponds to a $k$-cube. |
243 (This correspondence works best if we think of each twig label $u_i$ as having the form |
243 (This correspondence works best if we think of each twig label $u_i$ as having the form |
244 $x - s(e(x))$, where $x$ is an arbitrary field on $B_i$, $e: \cC(B_i) \to C$ is the evaluation map, |
244 $x - s(e(x))$, where $x$ is an arbitrary field on $B_i$, $e: \cF(B_i) \to C$ is the evaluation map, |
245 and $s:C \to \cC(B_i)$ is some fixed section of $e$. \todo{This parenthetical remark mysteriously specialises to the category case}) |
245 and $s:C \to \cF(B_i)$ is some fixed section of $e$. \todo{This parenthetical remark mysteriously specialises to the category case}) |
246 |
246 |
247 |
247 |