author | Kevin Walker <kevin@canyon23.net> |
Thu, 22 Jul 2010 19:32:40 -0600 | |
changeset 477 | 86c8e2129355 |
parent 474 | 6a3bc1c10586 |
child 479 | cfad13b6b1e5 |
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. |
|
437 | 7 |
Let $\cC$ be a fixed system of fields and local relations. |
472
0b9293d2f3f2
prelim blobdef.tex; committing now in anticipation Scott's further edits
Kevin Walker <kevin@canyon23.net>
parents:
466
diff
changeset
|
8 |
We'll assume it is enriched over \textbf{Vect}; |
0b9293d2f3f2
prelim blobdef.tex; committing now in anticipation Scott's further edits
Kevin Walker <kevin@canyon23.net>
parents:
466
diff
changeset
|
9 |
if it is not we can make it so by allowing finite |
437 | 10 |
linear combinations of elements of $\cC(X; c)$, for fixed $c\in \cC(\bd X)$. |
216 | 11 |
|
466
d238aa1ec317
fixed up much of blobdef, some examples and todos to deal with still. feel free to dismantle and reorganise\!
Scott Morrison <scott@tqft.net>
parents:
465
diff
changeset
|
12 |
%In this section we will usually suppress boundary conditions on $X$ from the notation, e.g. by writing $\lf(X)$ instead of $\lf(X; c)$. |
215 | 13 |
|
14 |
We want to replace the quotient |
|
15 |
\[ |
|
16 |
A(X) \deq \lf(X) / U(X) |
|
17 |
\] |
|
437 | 18 |
of Definition \ref{defn:TQFT-invariant} with a resolution |
215 | 19 |
\[ |
20 |
\cdots \to \bc_2(X) \to \bc_1(X) \to \bc_0(X) . |
|
21 |
\] |
|
22 |
||
466
d238aa1ec317
fixed up much of blobdef, some examples and todos to deal with still. feel free to dismantle and reorganise\!
Scott Morrison <scott@tqft.net>
parents:
465
diff
changeset
|
23 |
We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case $\bc_k(X)$. |
473 | 24 |
In fact, on the first pass we will intentionally describe the definition in a misleadingly simple way, |
25 |
then explain the technical difficulties, and finally give a cumbersome but complete definition in |
|
26 |
Definition \ref{defn:blobs}. |
|
27 |
If (we don't recommend it) you want to keep track of the ways in which |
|
28 |
this initial description is misleading, or you're reading through a second time to understand the |
|
29 |
technical difficulties, keep note that later we will give precise meanings to ``a ball in $X$'', |
|
472
0b9293d2f3f2
prelim blobdef.tex; committing now in anticipation Scott's further edits
Kevin Walker <kevin@canyon23.net>
parents:
466
diff
changeset
|
30 |
``nested'' and ``disjoint'', that are not quite the intuitive ones. |
473 | 31 |
Moreover some of the pieces |
32 |
into which we cut manifolds below are not themselves manifolds, and it requires special attention |
|
33 |
to define fields on these pieces. |
|
215 | 34 |
|
35 |
We of course define $\bc_0(X) = \lf(X)$. |
|
472
0b9293d2f3f2
prelim blobdef.tex; committing now in anticipation Scott's further edits
Kevin Walker <kevin@canyon23.net>
parents:
466
diff
changeset
|
36 |
(If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \lf(X; c)$ for each $c \in \lf(\bdy X)$. |
0b9293d2f3f2
prelim blobdef.tex; committing now in anticipation Scott's further edits
Kevin Walker <kevin@canyon23.net>
parents:
466
diff
changeset
|
37 |
We'll omit such boundary conditions from the notation in the rest of this section.) |
437 | 38 |
In other words, $\bc_0(X)$ is just the vector space of all fields on $X$. |
215 | 39 |
|
472
0b9293d2f3f2
prelim blobdef.tex; committing now in anticipation Scott's further edits
Kevin Walker <kevin@canyon23.net>
parents:
466
diff
changeset
|
40 |
We want the vector space $\bc_1(X)$ to capture |
0b9293d2f3f2
prelim blobdef.tex; committing now in anticipation Scott's further edits
Kevin Walker <kevin@canyon23.net>
parents:
466
diff
changeset
|
41 |
``the space of all local relations that can be imposed on $\bc_0(X)$". |
437 | 42 |
Thus we say a $1$-blob diagram consists of: |
215 | 43 |
\begin{itemize} |
466
d238aa1ec317
fixed up much of blobdef, some examples and todos to deal with still. feel free to dismantle and reorganise\!
Scott Morrison <scott@tqft.net>
parents:
465
diff
changeset
|
44 |
\item An closed ball in $X$ (``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
|
45 |
\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
|
46 |
\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
|
47 |
\item A local relation field $u \in U(B; c)$. |
215 | 48 |
\end{itemize} |
437 | 49 |
(See Figure \ref{blob1diagram}.) Since $c$ is implicitly determined by $u$ or $r$, we usually omit it from the notation. |
216 | 50 |
\begin{figure}[t]\begin{equation*} |
313 | 51 |
\mathfig{.6}{definition/single-blob} |
215 | 52 |
\end{equation*}\caption{A 1-blob diagram.}\label{blob1diagram}\end{figure} |
467
f3ccfc58ba7d
minor adjustments to blobdef
Scott Morrison <scott@tqft.net>
parents:
466
diff
changeset
|
53 |
In order to get the linear structure correct, we define |
215 | 54 |
\[ |
55 |
\bc_1(X) \deq \bigoplus_B \bigoplus_c U(B; c) \otimes \lf(X \setmin B; c) . |
|
56 |
\] |
|
57 |
The first direct sum is indexed by all blobs $B\subset X$, and the second |
|
58 |
by all boundary conditions $c \in \cC(\bd B)$. |
|
59 |
Note that $\bc_1(X)$ is spanned by 1-blob diagrams $(B, u, r)$. |
|
60 |
||
61 |
Define the boundary map $\bd : \bc_1(X) \to \bc_0(X)$ by |
|
62 |
\[ |
|
63 |
(B, u, r) \mapsto u\bullet r, |
|
64 |
\] |
|
216 | 65 |
where $u\bullet r$ denotes the field on $X$ obtained by gluing $u$ to $r$. |
215 | 66 |
In other words $\bd : \bc_1(X) \to \bc_0(X)$ is given by |
67 |
just erasing the blob from the picture |
|
68 |
(but keeping the blob label $u$). |
|
69 |
||
472
0b9293d2f3f2
prelim blobdef.tex; committing now in anticipation Scott's further edits
Kevin Walker <kevin@canyon23.net>
parents:
466
diff
changeset
|
70 |
\nn{it seems rather strange to make this a theorem} |
437 | 71 |
Note that directly from the definition we have |
72 |
\begin{thm} |
|
73 |
\label{thm:skein-modules} |
|
74 |
The skein module $A(X)$ is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$. |
|
75 |
\end{thm} |
|
76 |
This also establishes the second |
|
342 | 77 |
half of Property \ref{property:contractibility}. |
215 | 78 |
|
472
0b9293d2f3f2
prelim blobdef.tex; committing now in anticipation Scott's further edits
Kevin Walker <kevin@canyon23.net>
parents:
466
diff
changeset
|
79 |
Next, we want the vector space $\bc_2(X)$ to capture ``the space of all relations |
342 | 80 |
(redundancies, syzygies) among the |
472
0b9293d2f3f2
prelim blobdef.tex; committing now in anticipation Scott's further edits
Kevin Walker <kevin@canyon23.net>
parents:
466
diff
changeset
|
81 |
local relations encoded in $\bc_1(X)$''. |
437 | 82 |
A $2$-blob diagram, comes in one of two types, disjoint and nested. |
215 | 83 |
A disjoint 2-blob diagram consists of |
84 |
\begin{itemize} |
|
321
76c301fdf0a2
some changes to blobdef, in particular indexing starts at 1 now
Scott Morrison <scott@tqft.net>
parents:
313
diff
changeset
|
85 |
\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
|
86 |
\item A field $r \in \cC(X \setmin (B_1 \cup B_2); c_1, c_2)$ |
215 | 87 |
(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
|
88 |
\item Local relation fields $u_i \in U(B_i; c_i)$, $i=1,2$. |
215 | 89 |
\end{itemize} |
90 |
(See Figure \ref{blob2ddiagram}.) |
|
216 | 91 |
\begin{figure}[t]\begin{equation*} |
313 | 92 |
\mathfig{.6}{definition/disjoint-blobs} |
215 | 93 |
\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
|
94 |
We also identify $(B_1, B_2, u_1, u_2, r)$ with $-(B_2, B_1, u_2, u_1, r)$; |
215 | 95 |
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
|
96 |
Define $\bd(B_1, B_2, u_1, u_2, r) = |
76c301fdf0a2
some changes to blobdef, in particular indexing starts at 1 now
Scott Morrison <scott@tqft.net>
parents:
313
diff
changeset
|
97 |
(B_2, u_2, u_1\bullet r) - (B_1, u_1, u_2\bullet r) \in \bc_1(X)$. |
215 | 98 |
In other words, the boundary of a disjoint 2-blob diagram |
99 |
is the sum (with alternating signs) |
|
100 |
of the two ways of erasing one of the blobs. |
|
101 |
It's easy to check that $\bd^2 = 0$. |
|
102 |
||
103 |
A nested 2-blob diagram consists of |
|
104 |
\begin{itemize} |
|
413 | 105 |
\item A pair of nested balls (blobs) $B_1 \subseteq B_2 \subseteq X$. |
342 | 106 |
\item A field $r' \in \cC(B_2 \setminus B_1; c_1, c_2)$ |
107 |
(for some $c_1 \in \cC(\bdy B_1)$ and $c_2 \in \cC(\bdy B_2)$). |
|
321
76c301fdf0a2
some changes to blobdef, in particular indexing starts at 1 now
Scott Morrison <scott@tqft.net>
parents:
313
diff
changeset
|
108 |
\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
|
109 |
\item A local relation field $u \in U(B_1; c_1)$. |
215 | 110 |
\end{itemize} |
111 |
(See Figure \ref{blob2ndiagram}.) |
|
216 | 112 |
\begin{figure}[t]\begin{equation*} |
313 | 113 |
\mathfig{.6}{definition/nested-blobs} |
215 | 114 |
\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
|
115 |
Define $\bd(B_1, B_2, u, r', r) = (B_2, u\bullet r', r) - (B_1, u, r' \bullet r)$. |
215 | 116 |
As in the disjoint 2-blob case, the boundary of a nested 2-blob is the alternating |
117 |
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
|
118 |
When we erase the inner blob, the outer blob inherits the label $u\bullet r'$. |
437 | 119 |
It is again easy to check that $\bd^2 = 0$. Note that the requirement that |
120 |
local relations are an ideal with respect to gluing guarantees that $u\bullet r' \in U(B_2)$. |
|
215 | 121 |
|
321
76c301fdf0a2
some changes to blobdef, in particular indexing starts at 1 now
Scott Morrison <scott@tqft.net>
parents:
313
diff
changeset
|
122 |
As with the $1$-blob diagrams, in order to get the linear structure correct the actual definition is |
215 | 123 |
\begin{eqnarray*} |
124 |
\bc_2(X) & \deq & |
|
125 |
\left( |
|
413 | 126 |
\bigoplus_{B_1, B_2\; \text{disjoint}} \bigoplus_{c_1, c_2} |
321
76c301fdf0a2
some changes to blobdef, in particular indexing starts at 1 now
Scott Morrison <scott@tqft.net>
parents:
313
diff
changeset
|
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) |
413 | 128 |
\right) \bigoplus \\ |
129 |
&& \quad\quad \left( |
|
321
76c301fdf0a2
some changes to blobdef, in particular indexing starts at 1 now
Scott Morrison <scott@tqft.net>
parents:
313
diff
changeset
|
130 |
\bigoplus_{B_1 \subset B_2} \bigoplus_{c_1, c_2} |
472
0b9293d2f3f2
prelim blobdef.tex; committing now in anticipation Scott's further edits
Kevin Walker <kevin@canyon23.net>
parents:
466
diff
changeset
|
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) |
215 | 132 |
\right) . |
133 |
\end{eqnarray*} |
|
472
0b9293d2f3f2
prelim blobdef.tex; committing now in anticipation Scott's further edits
Kevin Walker <kevin@canyon23.net>
parents:
466
diff
changeset
|
134 |
% __ (already said this above) |
0b9293d2f3f2
prelim blobdef.tex; committing now in anticipation Scott's further edits
Kevin Walker <kevin@canyon23.net>
parents:
466
diff
changeset
|
135 |
%For the disjoint blobs, reversing the ordering of $B_1$ and $B_2$ introduces a minus sign |
0b9293d2f3f2
prelim blobdef.tex; committing now in anticipation Scott's further edits
Kevin Walker <kevin@canyon23.net>
parents:
466
diff
changeset
|
136 |
%(rather than a new, linearly independent, 2-blob diagram). |
0b9293d2f3f2
prelim blobdef.tex; committing now in anticipation Scott's further edits
Kevin Walker <kevin@canyon23.net>
parents:
466
diff
changeset
|
137 |
|
474 | 138 |
\medskip |
139 |
||
140 |
Roughly, $\bc_k(X)$ is generated by configurations of $k$ blobs, pairwise disjoint or nested. |
|
141 |
The boundary is the alternating sum of erasing one of the blobs. |
|
142 |
In order to describe this general case in full detail, we must give a more precise description of |
|
143 |
which configurations of balls inside $X$ we permit. |
|
144 |
These configurations are generated by two operations: |
|
145 |
\begin{itemize} |
|
146 |
\item For any (possibly empty) configuration of blobs on an $n$-ball $D$, we can add |
|
147 |
$D$ itself as an outermost blob. |
|
148 |
(This is used in the proof of Proposition \ref{bcontract}.) |
|
149 |
\item If $X'$ is obtained from $X$ by gluing, then any permissible configuration of blobs |
|
150 |
on $X$ gives rise to a permissible configuration on $X'$. |
|
151 |
(This is necessary for Proposition \ref{blob-gluing}.) |
|
152 |
\end{itemize} |
|
153 |
Combining these two operations can give rise to configurations of blobs whose complement in $X$ is not |
|
154 |
a manifold. |
|
155 |
Thus will need to be more careful when speaking of a field $r$ on the complement of the blobs. |
|
156 |
||
157 |
||
158 |
%In order to precisely state the general definition, we'll need a suitable notion of cutting up a manifold into balls. |
|
159 |
\begin{defn} |
|
160 |
\label{defn:gluing-decomposition} |
|
161 |
A \emph{gluing decomposition} of an $n$-manifold $X$ is a sequence of manifolds |
|
162 |
$M_0 \to M_1 \to \cdots \to M_m = X$ such that each $M_k$ is obtained from $M_{k-1}$ |
|
163 |
by gluing together some disjoint pair of homeomorphic $n{-}1$-manifolds in the boundary of $M_{k-1}$. |
|
164 |
If, in addition, $M_0$ is a disjoint union of balls, we call it a \emph{ball decomposition}. |
|
165 |
\end{defn} |
|
166 |
Given a gluing decomposition $M_0 \to M_1 \to \cdots \to M_m = X$, we say that a field is splittable along it if it is the image of a field on $M_0$. |
|
167 |
||
168 |
By ``a ball in $X$'' we don't literally mean a submanifold homeomorphic to a ball, but rather the image of a map from the pair $(B^n, S^{n-1})$ into $X$, which is an embedding on the interior. The boundary of a ball in $X$ is the image of a locally embedded $n{-}1$-sphere. \todo{examples, e.g. balls which actually look like an annulus, but we remember the boundary} |
|
169 |
\nn{not all balls in $X$ can arise via gluing, but I suppose that's OK.} |
|
170 |
||
171 |
\nn{do we need this next def?} |
|
172 |
\begin{defn} |
|
173 |
\label{defn:ball-decomposition} |
|
174 |
A \emph{ball decomposition} of an $n$-manifold $X$ is a collection of balls in $X$, such that there exists some gluing decomposition $M_0 \to \cdots \to M_m = X$ so that the balls are the images of the components of $M_0$ in $X$. |
|
175 |
\end{defn} |
|
176 |
In particular, the union of all the balls in a ball decomposition comprises all of $X$. \todo{example} |
|
177 |
||
178 |
We'll now slightly restrict the possible configurations of blobs. |
|
179 |
\begin{defn} |
|
180 |
\label{defn:configuration} |
|
181 |
A configuration of $k$ blobs in $X$ is an ordered collection of $k$ balls in $X$ such that there is some gluing decomposition $M_0 \to \cdots \to M_m = X$ of $X$ and each of the balls is the image of some connected component of one of the $M_k$. Such a gluing decomposition is \emph{compatible} with the configuration. |
|
182 |
\end{defn} |
|
183 |
In particular, this implies what we said about blobs above: |
|
184 |
that for any two blobs in a configuration of blobs in $X$, |
|
185 |
they either have disjoint interiors, or one blob is contained in the other. |
|
186 |
We describe these as disjoint blobs and nested blobs. |
|
187 |
Note that nested blobs may have boundaries that overlap, or indeed coincide. |
|
188 |
Blobs may meet the boundary of $X$. |
|
189 |
||
190 |
% (already said above) |
|
191 |
%Note that the boundaries of a configuration of $k$ blobs may cut up the manifold $X$ into components which are not themselves manifolds. \todo{example: the components between the boundaries of the balls may be pathological} |
|
192 |
||
193 |
In the informal description above, in the definition of a $k$-blob diagram we asked for any collection of $k$ balls which were pairwise disjoint or nested. We now further insist that the balls are a configuration in the sense of Definition \ref{defn:configuration}. Also, we asked for a local relation on each twig blob, and a field on the complement of the twig blobs; this is unsatisfactory because that complement need not be a manifold. Thus, the official definition is |
|
194 |
\begin{defn} |
|
195 |
\label{defn:blob-diagram} |
|
196 |
A $k$-blob diagram on $X$ consists of |
|
197 |
\begin{itemize} |
|
198 |
\item a configuration of $k$ blobs in $X$, |
|
199 |
\item and a field $r \in \cC(X)$ which is splittable along some gluing decomposition compatible with that configuration, |
|
200 |
\end{itemize} |
|
201 |
such that |
|
202 |
the restriction of $r$ to each twig blob $B_i$ lies in the subspace $U(B_i) \subset \cC(B_i)$. |
|
203 |
\end{defn} |
|
204 |
and |
|
205 |
\begin{defn} |
|
206 |
\label{defn:blobs} |
|
207 |
The $k$-th vector space $\bc_k(X)$ of the \emph{blob complex} of $X$ is the direct sum of all configurations of $k$ blobs in $X$ of the vector space of $k$-blob diagrams with that configuration, modulo identifying the vector spaces for configurations that only differ by a permutation of the balls by the sign of that permutation. The differential $bc_k(X) \to bc_{k-1}(X)$ is, as above, the signed sum of ways of forgetting one ball from the configuration, preserving the field $r$. |
|
208 |
\end{defn} |
|
209 |
We readily see that if a gluing decomposition is compatible with some configuration of blobs, then it is also compatible with any configuration obtained by forgetting some blobs, ensuring that the differential in fact lands in the space of $k{-}1$-blob diagrams. |
|
210 |
A slight compensation to the complication of the official definition arising from attention to splitting is that the differential now just preserves the entire field $r$ without having to say anything about gluing together fields on smaller components. |
|
472
0b9293d2f3f2
prelim blobdef.tex; committing now in anticipation Scott's further edits
Kevin Walker <kevin@canyon23.net>
parents:
466
diff
changeset
|
211 |
|
0b9293d2f3f2
prelim blobdef.tex; committing now in anticipation Scott's further edits
Kevin Walker <kevin@canyon23.net>
parents:
466
diff
changeset
|
212 |
|
413 | 213 |
|
474 | 214 |
|
215 |
||
216 |
||
217 |
||
218 |
\nn{should merge this informal def with official one above} |
|
219 |
||
220 |
Before describing the general case, note that when we say blobs are disjoint, |
|
221 |
we will only mean that their interiors are disjoint. |
|
222 |
Nested blobs may have boundaries that overlap, or indeed may coincide. |
|
215 | 223 |
A $k$-blob diagram consists of |
224 |
\begin{itemize} |
|
321
76c301fdf0a2
some changes to blobdef, in particular indexing starts at 1 now
Scott Morrison <scott@tqft.net>
parents:
313
diff
changeset
|
225 |
\item A collection of blobs $B_i \sub X$, $i = 1, \ldots, k$. |
216 | 226 |
For each $i$ and $j$, we require that either $B_i$ and $B_j$ have disjoint interiors or |
215 | 227 |
$B_i \sub B_j$ or $B_j \sub B_i$. |
228 |
If a blob has no other blobs strictly contained in it, we call it a twig blob. |
|
229 |
\item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$. |
|
230 |
(These are implied by the data in the next bullets, so we usually |
|
231 |
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
|
232 |
The fields $c_i$ and $c_j$ must have identical restrictions to $\bd B_i \cap \bd B_j$ |
215 | 233 |
if the latter space is not empty. |
234 |
\item A field $r \in \cC(X \setmin B^t; c^t)$, |
|
235 |
where $B^t$ is the union of all the twig blobs and $c^t \in \cC(\bd B^t)$ |
|
236 |
is determined by the $c_i$'s. |
|
474 | 237 |
The field $r$ is required to be splittable along the boundaries of all blobs, twigs or not. |
238 |
(This is equivalent to asking for a field on of the components of $X \setmin B^t$.) |
|
467
f3ccfc58ba7d
minor adjustments to blobdef
Scott Morrison <scott@tqft.net>
parents:
466
diff
changeset
|
239 |
\item For each twig blob $B_j$ a local relation field $u_j \in U(B_j; c_j)$. |
215 | 240 |
If $B_i = B_j$ then $u_i = u_j$. |
241 |
\end{itemize} |
|
242 |
(See Figure \ref{blobkdiagram}.) |
|
216 | 243 |
\begin{figure}[t]\begin{equation*} |
313 | 244 |
\mathfig{.7}{definition/k-blobs} |
215 | 245 |
\end{equation*}\caption{A $k$-blob diagram.}\label{blobkdiagram}\end{figure} |
246 |
||
247 |
If two blob diagrams $D_1$ and $D_2$ |
|
248 |
differ only by a reordering of the blobs, then we identify |
|
249 |
$D_1 = \pm D_2$, where the sign is the sign of the permutation relating $D_1$ and $D_2$. |
|
250 |
||
437 | 251 |
Roughly, then, $\bc_k(X)$ is all finite linear combinations of $k$-blob diagrams. |
215 | 252 |
As before, the official definition is in terms of direct sums |
253 |
of tensor products: |
|
254 |
\[ |
|
255 |
\bc_k(X) \deq \bigoplus_{\overline{B}} \bigoplus_{\overline{c}} |
|
437 | 256 |
\left( \bigotimes_j U(B_j; c_j)\right) \otimes \lf(X \setmin B^t; c^t) . |
215 | 257 |
\] |
258 |
Here $\overline{B}$ runs over all configurations of blobs, satisfying the conditions above. |
|
474 | 259 |
The index $\overline{c}$ runs over all boundary conditions, |
260 |
again as described above and $j$ runs over all indices of twig blobs. |
|
261 |
The final $\lf(X \setmin B^t; c^t)$ must be interpreted as fields which are |
|
262 |
splittable along all of the blobs in $\overline{B}$. |
|
215 | 263 |
|
216 | 264 |
The boundary map |
265 |
\[ |
|
266 |
\bd : \bc_k(X) \to \bc_{k-1}(X) |
|
267 |
\] |
|
268 |
is defined as follows. |
|
215 | 269 |
Let $b = (\{B_i\}, \{u_j\}, r)$ be a $k$-blob diagram. |
270 |
Let $E_j(b)$ denote the result of erasing the $j$-th blob. |
|
271 |
If $B_j$ is not a twig blob, this involves only decrementing |
|
437 | 272 |
the indices of blobs $B_{j+1},\ldots,B_{k}$. |
215 | 273 |
If $B_j$ is a twig blob, we have to assign new local relation labels |
474 | 274 |
if removing $B_j$ creates new twig blobs. |
275 |
\todo{Have to say what happens when no new twig blobs are created} |
|
276 |
\nn{KW: I'm confused --- why isn't it OK as written?} |
|
215 | 277 |
If $B_l$ becomes a twig after removing $B_j$, then set $u_l = u_j\bullet r_l$, |
278 |
where $r_l$ is the restriction of $r$ to $B_l \setmin B_j$. |
|
279 |
Finally, define |
|
280 |
\eq{ |
|
321
76c301fdf0a2
some changes to blobdef, in particular indexing starts at 1 now
Scott Morrison <scott@tqft.net>
parents:
313
diff
changeset
|
281 |
\bd(b) = \sum_{j=1}^{k} (-1)^{j+1} E_j(b). |
215 | 282 |
} |
321
76c301fdf0a2
some changes to blobdef, in particular indexing starts at 1 now
Scott Morrison <scott@tqft.net>
parents:
313
diff
changeset
|
283 |
The $(-1)^{j+1}$ factors imply that the terms of $\bd^2(b)$ all cancel. |
215 | 284 |
Thus we have a chain complex. |
285 |
||
474 | 286 |
Note that Property \ref{property:functoriality}, that the blob complex is functorial with respect to homeomorphisms, |
287 |
is immediately obvious from the definition. |
|
437 | 288 |
A homeomorphism acts in an obvious way on blobs and on fields. |
332
160ca7078ae9
fixing some inconsistencies in where the easy basic properties are treated
Scott Morrison <scott@tqft.net>
parents:
321
diff
changeset
|
289 |
|
466
d238aa1ec317
fixed up much of blobdef, some examples and todos to deal with still. feel free to dismantle and reorganise\!
Scott Morrison <scott@tqft.net>
parents:
465
diff
changeset
|
290 |
|
474 | 291 |
\nn{end relocated informal def} |
466
d238aa1ec317
fixed up much of blobdef, some examples and todos to deal with still. feel free to dismantle and reorganise\!
Scott Morrison <scott@tqft.net>
parents:
465
diff
changeset
|
292 |
|
d238aa1ec317
fixed up much of blobdef, some examples and todos to deal with still. feel free to dismantle and reorganise\!
Scott Morrison <scott@tqft.net>
parents:
465
diff
changeset
|
293 |
|
d238aa1ec317
fixed up much of blobdef, some examples and todos to deal with still. feel free to dismantle and reorganise\!
Scott Morrison <scott@tqft.net>
parents:
465
diff
changeset
|
294 |
|
d238aa1ec317
fixed up much of blobdef, some examples and todos to deal with still. feel free to dismantle and reorganise\!
Scott Morrison <scott@tqft.net>
parents:
465
diff
changeset
|
295 |
|
d238aa1ec317
fixed up much of blobdef, some examples and todos to deal with still. feel free to dismantle and reorganise\!
Scott Morrison <scott@tqft.net>
parents:
465
diff
changeset
|
296 |
|
257 | 297 |
We define the {\it support} of a blob diagram $b$, $\supp(b) \sub X$, |
298 |
to be the union of the blobs of $b$. |
|
299 |
For $y \in \bc_*(X)$ with $y = \sum c_i b_i$ ($c_i$ a non-zero number, $b_i$ a blob diagram), |
|
300 |
we define $\supp(y) \deq \bigcup_i \supp(b_i)$. |
|
301 |
||
216 | 302 |
We note that blob diagrams in $X$ have a structure similar to that of a simplicial set, |
303 |
but with simplices replaced by a more general class of combinatorial shapes. |
|
304 |
Let $P$ be the minimal set of (isomorphisms classes of) polyhedra which is closed under products |
|
305 |
and cones, and which contains the point. |
|
306 |
We can associate an element $p(b)$ of $P$ to each blob diagram $b$ |
|
307 |
(equivalently, to each rooted tree) according to the following rules: |
|
308 |
\begin{itemize} |
|
309 |
\item $p(\emptyset) = pt$, where $\emptyset$ denotes a 0-blob diagram or empty tree; |
|
342 | 310 |
\item $p(a \du b) = p(a) \times p(b)$, where $a \du b$ denotes the distant (non-overlapping) union |
311 |
of two blob diagrams (equivalently, join two trees at the roots); and |
|
312 |
\item $p(\bar{b}) = \kone(p(b))$, where $\bar{b}$ is obtained from $b$ by adding an outer blob which |
|
313 |
encloses all the others (equivalently, add a new edge to the root, with the new vertex becoming the root). |
|
216 | 314 |
\end{itemize} |
315 |
For example, a diagram of $k$ strictly nested blobs corresponds to a $k$-simplex, while |
|
316 |
a diagram of $k$ disjoint blobs corresponds to a $k$-cube. |
|
437 | 317 |
(This correspondence works best if we think of each twig label $u_i$ as having the form |
219 | 318 |
$x - s(e(x))$, where $x$ is an arbitrary field on $B_i$, $e: \cC(B_i) \to C$ is the evaluation map, |
437 | 319 |
and $s:C \to \cC(B_i)$ is some fixed section of $e$. \todo{This parenthetical remark mysteriously specialises to the category case}) |
215 | 320 |
|
321 |