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 (enriched over Vect) and local relations. |
7 Let $\cC$ be a fixed system of fields and local relations. |
8 (If $\cC$ is not enriched over Vect, we can make it so by allowing finite |
8 We'll assume it is enriched over \textbf{Vect}, and if it is not we can make it so by allowing finite |
9 linear combinations of elements of $\cC(X; c)$, for fixed $c\in \cC(\bd X)$.) |
9 linear combinations of elements of $\cC(X; c)$, for fixed $c\in \cC(\bd X)$. |
10 |
10 |
11 In this section we will usually suppress boundary conditions on $X$ from the notation |
11 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 (e.g. write $\lf(X)$ instead of $\lf(X; c)$). |
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13 |
12 |
14 We want to replace the quotient |
13 We want to replace the quotient |
15 \[ |
14 \[ |
16 A(X) \deq \lf(X) / U(X) |
15 A(X) \deq \lf(X) / U(X) |
17 \] |
16 \] |
18 of the previous section with a resolution |
17 of Definition \ref{defn:TQFT-invariant} with a resolution |
19 \[ |
18 \[ |
20 \cdots \to \bc_2(X) \to \bc_1(X) \to \bc_0(X) . |
19 \cdots \to \bc_2(X) \to \bc_1(X) \to \bc_0(X) . |
21 \] |
20 \] |
22 |
21 |
23 We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case $\bc_k(X)$. |
22 We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case $\bc_k(X)$. \todo{create a numbered definition for the general case} |
24 |
23 |
25 We of course define $\bc_0(X) = \lf(X)$. |
24 We of course define $\bc_0(X) = \lf(X)$. |
26 (If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \lf(X; c)$. |
25 (If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \lf(X; c)$. |
27 We'll omit this sort of detail in the rest of this section.) |
26 We'll omit this sort of detail in the rest of this section.) |
28 In other words, $\bc_0(X)$ is just the vector space of fields on $X$. |
27 In other words, $\bc_0(X)$ is just the vector space of all fields on $X$. |
29 |
28 |
30 We want the vector space $\bc_1(X)$ to capture `the space of all local relations that can be imposed on $\bc_0(X)$'. |
29 We want the vector space $\bc_1(X)$ to capture `the space of all local relations that can be imposed on $\bc_0(X)$'. |
31 Thus we say a $1$-blob diagram consists of |
30 Thus we say a $1$-blob diagram consists of: |
32 \begin{itemize} |
31 \begin{itemize} |
33 \item An embedded closed ball (``blob") $B \sub X$. |
32 \item An embedded closed ball (``blob") $B \sub X$. |
34 \item A boundary condition $c \in \cC(\bdy B) = \cC(\bd(X \setmin B))$. |
33 \item A boundary condition $c \in \cC(\bdy B) = \cC(\bd(X \setmin B))$. |
35 \item A field $r \in \cC(X \setmin B; c)$. |
34 \item A field $r \in \cC(X \setmin B; c)$. |
36 \item A local relation field $u \in U(B; c)$. |
35 \item A local relation field $u \in U(B; c)$. |
37 \end{itemize} |
36 \end{itemize} |
38 (See Figure \ref{blob1diagram}.) |
37 (See Figure \ref{blob1diagram}.) Since $c$ is implicitly determined by $u$ or $r$, we usually omit it from the notation. |
39 \begin{figure}[t]\begin{equation*} |
38 \begin{figure}[t]\begin{equation*} |
40 \mathfig{.6}{definition/single-blob} |
39 \mathfig{.6}{definition/single-blob} |
41 \end{equation*}\caption{A 1-blob diagram.}\label{blob1diagram}\end{figure} |
40 \end{equation*}\caption{A 1-blob diagram.}\label{blob1diagram}\end{figure} |
42 In order to get the linear structure correct, the actual definition is |
41 In order to get the linear structure correct, the actual definition is |
43 \[ |
42 \[ |
54 where $u\bullet r$ denotes the field on $X$ obtained by gluing $u$ to $r$. |
53 where $u\bullet r$ denotes the field on $X$ obtained by gluing $u$ to $r$. |
55 In other words $\bd : \bc_1(X) \to \bc_0(X)$ is given by |
54 In other words $\bd : \bc_1(X) \to \bc_0(X)$ is given by |
56 just erasing the blob from the picture |
55 just erasing the blob from the picture |
57 (but keeping the blob label $u$). |
56 (but keeping the blob label $u$). |
58 |
57 |
59 Note that the skein space $A(X)$ |
58 Note that directly from the definition we have |
60 is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$. |
59 \begin{thm} |
61 This is Theorem \ref{thm:skein-modules}, and also used in the second |
60 \label{thm:skein-modules} |
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61 The skein module $A(X)$ is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$. |
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62 \end{thm} |
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63 This also establishes the second |
62 half of Property \ref{property:contractibility}. |
64 half of Property \ref{property:contractibility}. |
63 |
65 |
64 Next, we want the vector space $\bc_2(X)$ to capture `the space of all relations |
66 Next, we want the vector space $\bc_2(X)$ to capture `the space of all relations |
65 (redundancies, syzygies) among the |
67 (redundancies, syzygies) among the |
66 local relations encoded in $\bc_1(X)$'. |
68 local relations encoded in $\bc_1(X)$'. |
67 More specifically, a $2$-blob diagram, comes in one of two types, disjoint and nested. |
69 A $2$-blob diagram, comes in one of two types, disjoint and nested. |
68 A disjoint 2-blob diagram consists of |
70 A disjoint 2-blob diagram consists of |
69 \begin{itemize} |
71 \begin{itemize} |
70 \item A pair of closed balls (blobs) $B_1, B_2 \sub X$ with disjoint interiors. |
72 \item A pair of closed balls (blobs) $B_1, B_2 \sub X$ with disjoint interiors. |
71 \item A field $r \in \cC(X \setmin (B_1 \cup B_2); c_1, c_2)$ |
73 \item A field $r \in \cC(X \setmin (B_1 \cup B_2); c_1, c_2)$ |
72 (where $c_i \in \cC(\bd B_i)$). |
74 (where $c_i \in \cC(\bd B_i)$). |
96 (See Figure \ref{blob2ndiagram}.) |
98 (See Figure \ref{blob2ndiagram}.) |
97 \begin{figure}[t]\begin{equation*} |
99 \begin{figure}[t]\begin{equation*} |
98 \mathfig{.6}{definition/nested-blobs} |
100 \mathfig{.6}{definition/nested-blobs} |
99 \end{equation*}\caption{A nested 2-blob diagram.}\label{blob2ndiagram}\end{figure} |
101 \end{equation*}\caption{A nested 2-blob diagram.}\label{blob2ndiagram}\end{figure} |
100 Define $\bd(B_1, B_2, u, r', r) = (B_2, u\bullet r', r) - (B_1, u, r' \bullet r)$. |
102 Define $\bd(B_1, B_2, u, r', r) = (B_2, u\bullet r', r) - (B_1, u, r' \bullet r)$. |
101 Note that the requirement that |
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102 local relations are an ideal with respect to gluing guarantees that $u\bullet r' \in U(B_2)$. |
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103 As in the disjoint 2-blob case, the boundary of a nested 2-blob is the alternating |
103 As in the disjoint 2-blob case, the boundary of a nested 2-blob is the alternating |
104 sum of the two ways of erasing one of the blobs. |
104 sum of the two ways of erasing one of the blobs. |
105 When we erase the inner blob, the outer blob inherits the label $u\bullet r'$. |
105 When we erase the inner blob, the outer blob inherits the label $u\bullet r'$. |
106 It is again easy to check that $\bd^2 = 0$. |
106 It is again easy to check that $\bd^2 = 0$. Note that the requirement that |
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107 local relations are an ideal with respect to gluing guarantees that $u\bullet r' \in U(B_2)$. |
107 |
108 |
108 As with the $1$-blob diagrams, in order to get the linear structure correct the actual definition is |
109 As with the $1$-blob diagrams, in order to get the linear structure correct the actual definition is |
109 \begin{eqnarray*} |
110 \begin{eqnarray*} |
110 \bc_2(X) & \deq & |
111 \bc_2(X) & \deq & |
111 \left( |
112 \left( |
115 && \quad\quad \left( |
116 && \quad\quad \left( |
116 \bigoplus_{B_1 \subset B_2} \bigoplus_{c_1, c_2} |
117 \bigoplus_{B_1 \subset B_2} \bigoplus_{c_1, c_2} |
117 U(B_1; c_1) \otimes \lf(B_2 \setmin B_1; c_1) \tensor \cC(X \setminus B_2; c_2) |
118 U(B_1; c_1) \otimes \lf(B_2 \setmin B_1; c_1) \tensor \cC(X \setminus B_2; c_2) |
118 \right) . |
119 \right) . |
119 \end{eqnarray*} |
120 \end{eqnarray*} |
120 For the disjoint blobs, reversing the ordering of $B_0$ and $B_1$ introduces a minus sign |
121 For the disjoint blobs, reversing the ordering of $B_1$ and $B_2$ introduces a minus sign |
121 (rather than a new, linearly independent 2-blob diagram). |
122 (rather than a new, linearly independent, 2-blob diagram). |
122 \noop{ |
123 \noop{ |
123 \nn{Hmm, I think we should be doing this for nested blobs too -- |
124 \nn{Hmm, I think we should be doing this for nested blobs too -- |
124 we shouldn't force the linear indexing of the blobs to have anything to do with |
125 we shouldn't force the linear indexing of the blobs to have anything to do with |
125 the partial ordering by inclusion -- this is what happens below} |
126 the partial ordering by inclusion -- this is what happens below} |
126 \nn{KW: I think adding that detail would only add distracting clutter, and the statement is true as written (in the sense that it yields a vector space isomorphic to the general def below} |
127 \nn{KW: I think adding that detail would only add distracting clutter, and the statement is true as written (in the sense that it yields a vector space isomorphic to the general def below} |
155 The fields $c_i$ and $c_j$ must have identical restrictions to $\bd B_i \cap \bd B_j$ |
156 The fields $c_i$ and $c_j$ must have identical restrictions to $\bd B_i \cap \bd B_j$ |
156 if the latter space is not empty. |
157 if the latter space is not empty. |
157 \item A field $r \in \cC(X \setmin B^t; c^t)$, |
158 \item A field $r \in \cC(X \setmin B^t; c^t)$, |
158 where $B^t$ is the union of all the twig blobs and $c^t \in \cC(\bd B^t)$ |
159 where $B^t$ is the union of all the twig blobs and $c^t \in \cC(\bd B^t)$ |
159 is determined by the $c_i$'s. |
160 is determined by the $c_i$'s. |
160 $r$ is required to be splittable along the boundaries of all blobs, twigs or not. |
161 $r$ is required to be splittable along the boundaries of all blobs, twigs or not. (This is equivalent to asking for a field on of the components of $X \setmin B^t$.) |
161 \item For each twig blob $B_j$ a local relation field $u_j \in U(B_j; c_j)$, |
162 \item For each twig blob $B_j$ a local relation field $u_j \in U(B_j; c_j)$, |
162 where $c_j$ is the restriction of $c^t$ to $\bd B_j$. |
163 where $c_j$ is the restriction of $c^t$ to $\bd B_j$. |
163 If $B_i = B_j$ then $u_i = u_j$. |
164 If $B_i = B_j$ then $u_i = u_j$. |
164 \end{itemize} |
165 \end{itemize} |
165 (See Figure \ref{blobkdiagram}.) |
166 (See Figure \ref{blobkdiagram}.) |
169 |
170 |
170 If two blob diagrams $D_1$ and $D_2$ |
171 If two blob diagrams $D_1$ and $D_2$ |
171 differ only by a reordering of the blobs, then we identify |
172 differ only by a reordering of the blobs, then we identify |
172 $D_1 = \pm D_2$, where the sign is the sign of the permutation relating $D_1$ and $D_2$. |
173 $D_1 = \pm D_2$, where the sign is the sign of the permutation relating $D_1$ and $D_2$. |
173 |
174 |
174 $\bc_k(X)$ is, roughly, all finite linear combinations of $k$-blob diagrams. |
175 Roughly, then, $\bc_k(X)$ is all finite linear combinations of $k$-blob diagrams. |
175 As before, the official definition is in terms of direct sums |
176 As before, the official definition is in terms of direct sums |
176 of tensor products: |
177 of tensor products: |
177 \[ |
178 \[ |
178 \bc_k(X) \deq \bigoplus_{\overline{B}} \bigoplus_{\overline{c}} |
179 \bc_k(X) \deq \bigoplus_{\overline{B}} \bigoplus_{\overline{c}} |
179 \left( \otimes_j U(B_j; c_j)\right) \otimes \lf(X \setmin B^t; c^t) . |
180 \left( \bigotimes_j U(B_j; c_j)\right) \otimes \lf(X \setmin B^t; c^t) . |
180 \] |
181 \] |
181 Here $\overline{B}$ runs over all configurations of blobs, satisfying the conditions above. |
182 Here $\overline{B}$ runs over all configurations of blobs, satisfying the conditions above. |
182 The index $\overline{c}$ runs over all boundary conditions, again as described above and $j$ runs over all indices of twig blobs. |
183 The index $\overline{c}$ runs over all boundary conditions, again as described above and $j$ runs over all indices of twig blobs. |
183 The final $\lf(X \setmin B^t; c^t)$ must be interpreted as fields which are splittable along all of the blobs in $\overline{B}$. |
184 The final $\lf(X \setmin B^t; c^t)$ must be interpreted as fields which are splittable along all of the blobs in $\overline{B}$. |
184 |
185 |
188 \] |
189 \] |
189 is defined as follows. |
190 is defined as follows. |
190 Let $b = (\{B_i\}, \{u_j\}, r)$ be a $k$-blob diagram. |
191 Let $b = (\{B_i\}, \{u_j\}, r)$ be a $k$-blob diagram. |
191 Let $E_j(b)$ denote the result of erasing the $j$-th blob. |
192 Let $E_j(b)$ denote the result of erasing the $j$-th blob. |
192 If $B_j$ is not a twig blob, this involves only decrementing |
193 If $B_j$ is not a twig blob, this involves only decrementing |
193 the indices of blobs $B_{j+1},\ldots,B_{k-1}$. |
194 the indices of blobs $B_{j+1},\ldots,B_{k}$. |
194 If $B_j$ is a twig blob, we have to assign new local relation labels |
195 If $B_j$ is a twig blob, we have to assign new local relation labels |
195 if removing $B_j$ creates new twig blobs. |
196 if removing $B_j$ creates new twig blobs. \todo{Have to say what happens when no new twig blobs are created} |
196 If $B_l$ becomes a twig after removing $B_j$, then set $u_l = u_j\bullet r_l$, |
197 If $B_l$ becomes a twig after removing $B_j$, then set $u_l = u_j\bullet r_l$, |
197 where $r_l$ is the restriction of $r$ to $B_l \setmin B_j$. |
198 where $r_l$ is the restriction of $r$ to $B_l \setmin B_j$. |
198 Finally, define |
199 Finally, define |
199 \eq{ |
200 \eq{ |
200 \bd(b) = \sum_{j=1}^{k} (-1)^{j+1} E_j(b). |
201 \bd(b) = \sum_{j=1}^{k} (-1)^{j+1} E_j(b). |
201 } |
202 } |
202 The $(-1)^{j+1}$ factors imply that the terms of $\bd^2(b)$ all cancel. |
203 The $(-1)^{j+1}$ factors imply that the terms of $\bd^2(b)$ all cancel. |
203 Thus we have a chain complex. |
204 Thus we have a chain complex. |
204 |
205 |
205 Note that Property \ref{property:functoriality}, that the blob complex is functorial with respect to homeomorphisms, is immediately obvious from the definition. |
206 Note that Property \ref{property:functoriality}, that the blob complex is functorial with respect to homeomorphisms, is immediately obvious from the definition. |
206 A homeomorphism acts in an obvious on blobs and on fields. |
207 A homeomorphism acts in an obvious way on blobs and on fields. |
207 |
208 |
208 We define the {\it support} of a blob diagram $b$, $\supp(b) \sub X$, |
209 We define the {\it support} of a blob diagram $b$, $\supp(b) \sub X$, |
209 to be the union of the blobs of $b$. |
210 to be the union of the blobs of $b$. |
210 For $y \in \bc_*(X)$ with $y = \sum c_i b_i$ ($c_i$ a non-zero number, $b_i$ a blob diagram), |
211 For $y \in \bc_*(X)$ with $y = \sum c_i b_i$ ($c_i$ a non-zero number, $b_i$ a blob diagram), |
211 we define $\supp(y) \deq \bigcup_i \supp(b_i)$. |
212 we define $\supp(y) \deq \bigcup_i \supp(b_i)$. |
223 \item $p(\bar{b}) = \kone(p(b))$, where $\bar{b}$ is obtained from $b$ by adding an outer blob which |
224 \item $p(\bar{b}) = \kone(p(b))$, where $\bar{b}$ is obtained from $b$ by adding an outer blob which |
224 encloses all the others (equivalently, add a new edge to the root, with the new vertex becoming the root). |
225 encloses all the others (equivalently, add a new edge to the root, with the new vertex becoming the root). |
225 \end{itemize} |
226 \end{itemize} |
226 For example, a diagram of $k$ strictly nested blobs corresponds to a $k$-simplex, while |
227 For example, a diagram of $k$ strictly nested blobs corresponds to a $k$-simplex, while |
227 a diagram of $k$ disjoint blobs corresponds to a $k$-cube. |
228 a diagram of $k$ disjoint blobs corresponds to a $k$-cube. |
228 (This correspondence works best if we thing of each twig label $u_i$ as having the form |
229 (This correspondence works best if we think of each twig label $u_i$ as having the form |
229 $x - s(e(x))$, where $x$ is an arbitrary field on $B_i$, $e: \cC(B_i) \to C$ is the evaluation map, |
230 $x - s(e(x))$, where $x$ is an arbitrary field on $B_i$, $e: \cC(B_i) \to C$ is the evaluation map, |
230 and $s:C \to \cC(B_i)$ is some fixed section of $e$.) |
231 and $s:C \to \cC(B_i)$ is some fixed section of $e$. \todo{This parenthetical remark mysteriously specialises to the category case}) |
231 |
232 |
232 |
233 |