author | Scott Morrison <scott@tqft.net> |
Thu, 03 Jun 2010 23:08:47 -0700 | |
changeset 338 | adc0780aa5e7 |
parent 332 | 160ca7078ae9 |
child 342 | 1d76e832d32f |
permissions | -rw-r--r-- |
215 | 1 |
%!TEX root = ../blob1.tex |
2 |
||
3 |
\section{The blob complex} |
|
4 |
\label{sec:blob-definition} |
|
5 |
||
6 |
Let $X$ be an $n$-manifold. |
|
216 | 7 |
Let $\cC$ be a fixed system of fields (enriched over Vect) and local relations. |
8 |
(If $\cC$ is not enriched over Vect, we can make it so by allowing finite |
|
9 |
linear combinations of elements of $\cC(X; c)$, for fixed $c\in \cC(\bd X)$.) |
|
10 |
||
215 | 11 |
In this section we will usually suppress boundary conditions on $X$ from the notation |
12 |
(e.g. write $\lf(X)$ instead of $\lf(X; c)$). |
|
13 |
||
14 |
We want to replace the quotient |
|
15 |
\[ |
|
16 |
A(X) \deq \lf(X) / U(X) |
|
17 |
\] |
|
18 |
of the previous section with a resolution |
|
19 |
\[ |
|
20 |
\cdots \to \bc_2(X) \to \bc_1(X) \to \bc_0(X) . |
|
21 |
\] |
|
22 |
||
23 |
We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case $\bc_k(X)$. |
|
24 |
||
25 |
We of course define $\bc_0(X) = \lf(X)$. |
|
26 |
(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.) |
|
321
76c301fdf0a2
some changes to blobdef, in particular indexing starts at 1 now
Scott Morrison <scott@tqft.net>
parents:
313
diff
changeset
|
28 |
In other words, $\bc_0(X)$ is just the vector space of fields on $X$. |
215 | 29 |
|
321
76c301fdf0a2
some changes to blobdef, in particular indexing starts at 1 now
Scott Morrison <scott@tqft.net>
parents:
313
diff
changeset
|
30 |
We want the vector space $\bc_1(X)$ to capture `the space of all local relations that can be imposed on $\bc_0(X)$'. |
76c301fdf0a2
some changes to blobdef, in particular indexing starts at 1 now
Scott Morrison <scott@tqft.net>
parents:
313
diff
changeset
|
31 |
Thus we say a $1$-blob diagram consists of |
215 | 32 |
\begin{itemize} |
33 |
\item An embedded closed ball (``blob") $B \sub X$. |
|
321
76c301fdf0a2
some changes to blobdef, in particular indexing starts at 1 now
Scott Morrison <scott@tqft.net>
parents:
313
diff
changeset
|
34 |
\item A boundary condition $c \in \cC(\bdy B) = \cC(\bd(X \setmin B))$. |
76c301fdf0a2
some changes to blobdef, in particular indexing starts at 1 now
Scott Morrison <scott@tqft.net>
parents:
313
diff
changeset
|
35 |
\item A field $r \in \cC(X \setmin B; c)$. |
76c301fdf0a2
some changes to blobdef, in particular indexing starts at 1 now
Scott Morrison <scott@tqft.net>
parents:
313
diff
changeset
|
36 |
\item A local relation field $u \in U(B; c)$. |
215 | 37 |
\end{itemize} |
38 |
(See Figure \ref{blob1diagram}.) |
|
216 | 39 |
\begin{figure}[t]\begin{equation*} |
313 | 40 |
\mathfig{.6}{definition/single-blob} |
215 | 41 |
\end{equation*}\caption{A 1-blob diagram.}\label{blob1diagram}\end{figure} |
321
76c301fdf0a2
some changes to blobdef, in particular indexing starts at 1 now
Scott Morrison <scott@tqft.net>
parents:
313
diff
changeset
|
42 |
In order to get the linear structure correct, the actual definition is |
215 | 43 |
\[ |
44 |
\bc_1(X) \deq \bigoplus_B \bigoplus_c U(B; c) \otimes \lf(X \setmin B; c) . |
|
45 |
\] |
|
46 |
The first direct sum is indexed by all blobs $B\subset X$, and the second |
|
47 |
by all boundary conditions $c \in \cC(\bd B)$. |
|
48 |
Note that $\bc_1(X)$ is spanned by 1-blob diagrams $(B, u, r)$. |
|
49 |
||
50 |
Define the boundary map $\bd : \bc_1(X) \to \bc_0(X)$ by |
|
51 |
\[ |
|
52 |
(B, u, r) \mapsto u\bullet r, |
|
53 |
\] |
|
216 | 54 |
where $u\bullet r$ denotes the field on $X$ obtained by gluing $u$ to $r$. |
215 | 55 |
In other words $\bd : \bc_1(X) \to \bc_0(X)$ is given by |
56 |
just erasing the blob from the picture |
|
57 |
(but keeping the blob label $u$). |
|
58 |
||
59 |
Note that the skein space $A(X)$ |
|
222
217b6a870532
committing changes from loon lake - mostly small blobs
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
219
diff
changeset
|
60 |
is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$. This is Property \ref{property:skein-modules}, and also used in the second half of Property \ref{property:contractibility}. |
215 | 61 |
|
321
76c301fdf0a2
some changes to blobdef, in particular indexing starts at 1 now
Scott Morrison <scott@tqft.net>
parents:
313
diff
changeset
|
62 |
Next, we want the vector space $\bc_2(X)$ to capture `the space of all relations (redundancies, syzygies) among the |
76c301fdf0a2
some changes to blobdef, in particular indexing starts at 1 now
Scott Morrison <scott@tqft.net>
parents:
313
diff
changeset
|
63 |
local relations encoded in $\bc_1(X)$'. |
76c301fdf0a2
some changes to blobdef, in particular indexing starts at 1 now
Scott Morrison <scott@tqft.net>
parents:
313
diff
changeset
|
64 |
More specifically, a $2$-blob diagram, comes in one of two types, disjoint and nested. |
215 | 65 |
A disjoint 2-blob diagram consists of |
66 |
\begin{itemize} |
|
321
76c301fdf0a2
some changes to blobdef, in particular indexing starts at 1 now
Scott Morrison <scott@tqft.net>
parents:
313
diff
changeset
|
67 |
\item A pair of closed balls (blobs) $B_1, B_2 \sub X$ with disjoint interiors. |
76c301fdf0a2
some changes to blobdef, in particular indexing starts at 1 now
Scott Morrison <scott@tqft.net>
parents:
313
diff
changeset
|
68 |
\item A field $r \in \cC(X \setmin (B_1 \cup B_2); c_1, c_2)$ |
215 | 69 |
(where $c_i \in \cC(\bd B_i)$). |
321
76c301fdf0a2
some changes to blobdef, in particular indexing starts at 1 now
Scott Morrison <scott@tqft.net>
parents:
313
diff
changeset
|
70 |
\item Local relation fields $u_i \in U(B_i; c_i)$, $i=1,2$. |
215 | 71 |
\end{itemize} |
72 |
(See Figure \ref{blob2ddiagram}.) |
|
216 | 73 |
\begin{figure}[t]\begin{equation*} |
313 | 74 |
\mathfig{.6}{definition/disjoint-blobs} |
215 | 75 |
\end{equation*}\caption{A disjoint 2-blob diagram.}\label{blob2ddiagram}\end{figure} |
321
76c301fdf0a2
some changes to blobdef, in particular indexing starts at 1 now
Scott Morrison <scott@tqft.net>
parents:
313
diff
changeset
|
76 |
We also identify $(B_1, B_2, u_1, u_2, r)$ with $-(B_2, B_1, u_2, u_1, r)$; |
215 | 77 |
reversing the order of the blobs changes the sign. |
321
76c301fdf0a2
some changes to blobdef, in particular indexing starts at 1 now
Scott Morrison <scott@tqft.net>
parents:
313
diff
changeset
|
78 |
Define $\bd(B_1, B_2, u_1, u_2, r) = |
76c301fdf0a2
some changes to blobdef, in particular indexing starts at 1 now
Scott Morrison <scott@tqft.net>
parents:
313
diff
changeset
|
79 |
(B_2, u_2, u_1\bullet r) - (B_1, u_1, u_2\bullet r) \in \bc_1(X)$. |
215 | 80 |
In other words, the boundary of a disjoint 2-blob diagram |
81 |
is the sum (with alternating signs) |
|
82 |
of the two ways of erasing one of the blobs. |
|
83 |
It's easy to check that $\bd^2 = 0$. |
|
84 |
||
85 |
A nested 2-blob diagram consists of |
|
86 |
\begin{itemize} |
|
321
76c301fdf0a2
some changes to blobdef, in particular indexing starts at 1 now
Scott Morrison <scott@tqft.net>
parents:
313
diff
changeset
|
87 |
\item A pair of nested balls (blobs) $B_1 \sub B_2 \sub X$. |
76c301fdf0a2
some changes to blobdef, in particular indexing starts at 1 now
Scott Morrison <scott@tqft.net>
parents:
313
diff
changeset
|
88 |
\item A field $r' \in \cC(B_2 \setminus B_1; c_1, c_2)$ (for some $c_1 \in \cC(\bdy B_1)$ and $c_2 \in \cC(\bdy B_2)$). |
76c301fdf0a2
some changes to blobdef, in particular indexing starts at 1 now
Scott Morrison <scott@tqft.net>
parents:
313
diff
changeset
|
89 |
\item A field $r \in \cC(X \setminus B_2; c_2)$. |
76c301fdf0a2
some changes to blobdef, in particular indexing starts at 1 now
Scott Morrison <scott@tqft.net>
parents:
313
diff
changeset
|
90 |
\item A local relation field $u \in U(B_1; c_1)$. |
215 | 91 |
\end{itemize} |
92 |
(See Figure \ref{blob2ndiagram}.) |
|
216 | 93 |
\begin{figure}[t]\begin{equation*} |
313 | 94 |
\mathfig{.6}{definition/nested-blobs} |
215 | 95 |
\end{equation*}\caption{A nested 2-blob diagram.}\label{blob2ndiagram}\end{figure} |
321
76c301fdf0a2
some changes to blobdef, in particular indexing starts at 1 now
Scott Morrison <scott@tqft.net>
parents:
313
diff
changeset
|
96 |
Define $\bd(B_1, B_2, u, r', r) = (B_2, u\bullet r', r) - (B_1, u, r' \bullet r)$. |
215 | 97 |
Note that the requirement that |
321
76c301fdf0a2
some changes to blobdef, in particular indexing starts at 1 now
Scott Morrison <scott@tqft.net>
parents:
313
diff
changeset
|
98 |
local relations are an ideal with respect to gluing guarantees that $u\bullet r' \in U(B_2)$. |
215 | 99 |
As in the disjoint 2-blob case, the boundary of a nested 2-blob is the alternating |
100 |
sum of the two ways of erasing one of the blobs. |
|
321
76c301fdf0a2
some changes to blobdef, in particular indexing starts at 1 now
Scott Morrison <scott@tqft.net>
parents:
313
diff
changeset
|
101 |
When we erase the inner blob, the outer blob inherits the label $u\bullet r'$. |
215 | 102 |
It is again easy to check that $\bd^2 = 0$. |
103 |
||
321
76c301fdf0a2
some changes to blobdef, in particular indexing starts at 1 now
Scott Morrison <scott@tqft.net>
parents:
313
diff
changeset
|
104 |
As with the $1$-blob diagrams, in order to get the linear structure correct the actual definition is |
215 | 105 |
\begin{eqnarray*} |
106 |
\bc_2(X) & \deq & |
|
107 |
\left( |
|
321
76c301fdf0a2
some changes to blobdef, in particular indexing starts at 1 now
Scott Morrison <scott@tqft.net>
parents:
313
diff
changeset
|
108 |
\bigoplus_{B_1, B_2 \text{disjoint}} \bigoplus_{c_1, c_2} |
76c301fdf0a2
some changes to blobdef, in particular indexing starts at 1 now
Scott Morrison <scott@tqft.net>
parents:
313
diff
changeset
|
109 |
U(B_1; c_1) \otimes U(B_2; c_2) \otimes \lf(X\setmin (B_1\cup B_2); c_1, c_2) |
215 | 110 |
\right) \\ |
111 |
&& \bigoplus \left( |
|
321
76c301fdf0a2
some changes to blobdef, in particular indexing starts at 1 now
Scott Morrison <scott@tqft.net>
parents:
313
diff
changeset
|
112 |
\bigoplus_{B_1 \subset B_2} \bigoplus_{c_1, c_2} |
76c301fdf0a2
some changes to blobdef, in particular indexing starts at 1 now
Scott Morrison <scott@tqft.net>
parents:
313
diff
changeset
|
113 |
U(B_1; c_1) \otimes \lf(B_2 \setmin B_1; c_1) \tensor \cC(X \setminus B_2; c_2) |
215 | 114 |
\right) . |
115 |
\end{eqnarray*} |
|
116 |
For the disjoint blobs, reversing the ordering of $B_0$ and $B_1$ introduces a minus sign |
|
321
76c301fdf0a2
some changes to blobdef, in particular indexing starts at 1 now
Scott Morrison <scott@tqft.net>
parents:
313
diff
changeset
|
117 |
(rather than a new, linearly independent 2-blob diagram). \nn{Hmm, I think we should be doing this for nested blobs too -- we shouldn't force the linear indexing of the blobs to have anything to do with the partial ordering by inclusion -- this is what happens below} |
215 | 118 |
|
119 |
Now for the general case. |
|
120 |
A $k$-blob diagram consists of |
|
121 |
\begin{itemize} |
|
321
76c301fdf0a2
some changes to blobdef, in particular indexing starts at 1 now
Scott Morrison <scott@tqft.net>
parents:
313
diff
changeset
|
122 |
\item A collection of blobs $B_i \sub X$, $i = 1, \ldots, k$. |
216 | 123 |
For each $i$ and $j$, we require that either $B_i$ and $B_j$ have disjoint interiors or |
215 | 124 |
$B_i \sub B_j$ or $B_j \sub B_i$. |
125 |
(The case $B_i = B_j$ is allowed. |
|
126 |
If $B_i \sub B_j$ the boundaries of $B_i$ and $B_j$ are allowed to intersect.) |
|
127 |
If a blob has no other blobs strictly contained in it, we call it a twig blob. |
|
257 | 128 |
\nn{need to allow the case where $B\to X$ is not an embedding |
129 |
on $\bd B$. this is because any blob diagram on $X_{cut}$ should give rise to one on $X_{gl}$ |
|
130 |
and blobs are allowed to meet $\bd X$.} |
|
215 | 131 |
\item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$. |
132 |
(These are implied by the data in the next bullets, so we usually |
|
133 |
suppress them from the notation.) |
|
321
76c301fdf0a2
some changes to blobdef, in particular indexing starts at 1 now
Scott Morrison <scott@tqft.net>
parents:
313
diff
changeset
|
134 |
The fields $c_i$ and $c_j$ must have identical restrictions to $\bd B_i \cap \bd B_j$ |
215 | 135 |
if the latter space is not empty. |
136 |
\item A field $r \in \cC(X \setmin B^t; c^t)$, |
|
137 |
where $B^t$ is the union of all the twig blobs and $c^t \in \cC(\bd B^t)$ |
|
138 |
is determined by the $c_i$'s. |
|
139 |
$r$ is required to be splittable along the boundaries of all blobs, twigs or not. |
|
140 |
\item For each twig blob $B_j$ a local relation field $u_j \in U(B_j; c_j)$, |
|
141 |
where $c_j$ is the restriction of $c^t$ to $\bd B_j$. |
|
142 |
If $B_i = B_j$ then $u_i = u_j$. |
|
143 |
\end{itemize} |
|
144 |
(See Figure \ref{blobkdiagram}.) |
|
216 | 145 |
\begin{figure}[t]\begin{equation*} |
313 | 146 |
\mathfig{.7}{definition/k-blobs} |
215 | 147 |
\end{equation*}\caption{A $k$-blob diagram.}\label{blobkdiagram}\end{figure} |
148 |
||
149 |
If two blob diagrams $D_1$ and $D_2$ |
|
150 |
differ only by a reordering of the blobs, then we identify |
|
151 |
$D_1 = \pm D_2$, where the sign is the sign of the permutation relating $D_1$ and $D_2$. |
|
152 |
||
153 |
$\bc_k(X)$ is, roughly, all finite linear combinations of $k$-blob diagrams. |
|
154 |
As before, the official definition is in terms of direct sums |
|
155 |
of tensor products: |
|
156 |
\[ |
|
157 |
\bc_k(X) \deq \bigoplus_{\overline{B}} \bigoplus_{\overline{c}} |
|
158 |
\left( \otimes_j U(B_j; c_j)\right) \otimes \lf(X \setmin B^t; c^t) . |
|
159 |
\] |
|
160 |
Here $\overline{B}$ runs over all configurations of blobs, satisfying the conditions above. |
|
321
76c301fdf0a2
some changes to blobdef, in particular indexing starts at 1 now
Scott Morrison <scott@tqft.net>
parents:
313
diff
changeset
|
161 |
The index $\overline{c}$ runs over all boundary conditions, again as described above and $j$ runs over all indices of twig blobs. The final $\lf(X \setmin B^t; c^t)$ must be interpreted as fields which are splittable along all of the blobs in $\overline{B}$. |
215 | 162 |
|
216 | 163 |
The boundary map |
164 |
\[ |
|
165 |
\bd : \bc_k(X) \to \bc_{k-1}(X) |
|
166 |
\] |
|
167 |
is defined as follows. |
|
215 | 168 |
Let $b = (\{B_i\}, \{u_j\}, r)$ be a $k$-blob diagram. |
169 |
Let $E_j(b)$ denote the result of erasing the $j$-th blob. |
|
170 |
If $B_j$ is not a twig blob, this involves only decrementing |
|
171 |
the indices of blobs $B_{j+1},\ldots,B_{k-1}$. |
|
172 |
If $B_j$ is a twig blob, we have to assign new local relation labels |
|
173 |
if removing $B_j$ creates new twig blobs. |
|
174 |
If $B_l$ becomes a twig after removing $B_j$, then set $u_l = u_j\bullet r_l$, |
|
175 |
where $r_l$ is the restriction of $r$ to $B_l \setmin B_j$. |
|
176 |
Finally, define |
|
177 |
\eq{ |
|
321
76c301fdf0a2
some changes to blobdef, in particular indexing starts at 1 now
Scott Morrison <scott@tqft.net>
parents:
313
diff
changeset
|
178 |
\bd(b) = \sum_{j=1}^{k} (-1)^{j+1} E_j(b). |
215 | 179 |
} |
321
76c301fdf0a2
some changes to blobdef, in particular indexing starts at 1 now
Scott Morrison <scott@tqft.net>
parents:
313
diff
changeset
|
180 |
The $(-1)^{j+1}$ factors imply that the terms of $\bd^2(b)$ all cancel. |
215 | 181 |
Thus we have a chain complex. |
182 |
||
332
160ca7078ae9
fixing some inconsistencies in where the easy basic properties are treated
Scott Morrison <scott@tqft.net>
parents:
321
diff
changeset
|
183 |
Note that Property \ref{property:functoriality}, that the blob complex is functorial with respect to homeomorphisms, is immediately obvious from the definition. A homeomorphism acts in an obvious on blobs and on fields. |
160ca7078ae9
fixing some inconsistencies in where the easy basic properties are treated
Scott Morrison <scott@tqft.net>
parents:
321
diff
changeset
|
184 |
|
257 | 185 |
We define the {\it support} of a blob diagram $b$, $\supp(b) \sub X$, |
186 |
to be the union of the blobs of $b$. |
|
187 |
For $y \in \bc_*(X)$ with $y = \sum c_i b_i$ ($c_i$ a non-zero number, $b_i$ a blob diagram), |
|
188 |
we define $\supp(y) \deq \bigcup_i \supp(b_i)$. |
|
189 |
||
216 | 190 |
We note that blob diagrams in $X$ have a structure similar to that of a simplicial set, |
191 |
but with simplices replaced by a more general class of combinatorial shapes. |
|
192 |
Let $P$ be the minimal set of (isomorphisms classes of) polyhedra which is closed under products |
|
193 |
and cones, and which contains the point. |
|
194 |
We can associate an element $p(b)$ of $P$ to each blob diagram $b$ |
|
195 |
(equivalently, to each rooted tree) according to the following rules: |
|
196 |
\begin{itemize} |
|
197 |
\item $p(\emptyset) = pt$, where $\emptyset$ denotes a 0-blob diagram or empty tree; |
|
198 |
\item $p(a \du b) = p(a) \times p(b)$, where $a \du b$ denotes the distant (non-overlapping) union of two blob diagrams (equivalently, join two trees at the roots); and |
|
321
76c301fdf0a2
some changes to blobdef, in particular indexing starts at 1 now
Scott Morrison <scott@tqft.net>
parents:
313
diff
changeset
|
199 |
\item $p(\bar{b}) = \kone(p(b))$, where $\bar{b}$ is obtained from $b$ by adding an outer blob which encloses all the others (equivalently, add a new edge to the root, with the new vertex becoming the root). |
216 | 200 |
\end{itemize} |
201 |
For example, a diagram of $k$ strictly nested blobs corresponds to a $k$-simplex, while |
|
202 |
a diagram of $k$ disjoint blobs corresponds to a $k$-cube. |
|
219 | 203 |
(This correspondence works best if we thing of each twig label $u_i$ as having the form |
204 |
$x - s(e(x))$, where $x$ is an arbitrary field on $B_i$, $e: \cC(B_i) \to C$ is the evaluation map, |
|
205 |
and $s:C \to \cC(B_i)$ is some fixed section of $e$.) |
|
215 | 206 |
|
207 |