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1 |
%!TEX root = ../blob1.tex
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2 |
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\section{The blob complex}
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4 |
\label{sec:blob-definition}
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5 |
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Let $X$ be an $n$-manifold.
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7 |
Let $\cC$ be a fixed system of fields (enriched over Vect) and local relations.
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|
8 |
(If $\cC$ is not enriched over Vect, we can make it so by allowing finite
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|
9 |
linear combinations of elements of $\cC(X; c)$, for fixed $c\in \cC(\bd X)$.)
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10 |
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11 |
In this section we will usually suppress boundary conditions on $X$ from the notation
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|
12 |
(e.g. write $\lf(X)$ instead of $\lf(X; c)$).
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13 |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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14 |
We want to replace the quotient
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|
15 |
\[
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parents:
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16 |
A(X) \deq \lf(X) / U(X)
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
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|
17 |
\]
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
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18 |
of the previous section with a resolution
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
19 |
\[
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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20 |
\cdots \to \bc_2(X) \to \bc_1(X) \to \bc_0(X) .
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
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21 |
\]
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
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22 |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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23 |
We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case $\bc_k(X)$.
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24 |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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25 |
We of course define $\bc_0(X) = \lf(X)$.
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|
26 |
(If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \lf(X; c)$.
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27 |
We'll omit this sort of detail in the rest of this section.)
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28 |
In other words, $\bc_0(X)$ is just the vector space of all (linearized) fields on $X$.
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29 |
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30 |
$\bc_1(X)$ is, roughly, the space of all local relations that can be imposed on $\bc_0(X)$.
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31 |
Less roughly (but still not the official definition), $\bc_1(X)$ is finite linear
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|
32 |
combinations of 1-blob diagrams, where a 1-blob diagram consists of
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33 |
\begin{itemize}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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34 |
\item An embedded closed ball (``blob") $B \sub X$.
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35 |
\item A field $r \in \cC(X \setmin B; c)$
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36 |
(for some $c \in \cC(\bd B) = \cC(\bd(X \setmin B))$).
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37 |
\item A local relation field $u \in U(B; c)$
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|
38 |
(same $c$ as previous bullet).
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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39 |
\end{itemize}
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40 |
(See Figure \ref{blob1diagram}.)
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41 |
\begin{figure}[t]\begin{equation*}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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42 |
\mathfig{.9}{definition/single-blob}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
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|
43 |
\end{equation*}\caption{A 1-blob diagram.}\label{blob1diagram}\end{figure}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
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|
44 |
In order to get the linear structure correct, we (officially) define
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
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|
45 |
\[
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
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|
46 |
\bc_1(X) \deq \bigoplus_B \bigoplus_c U(B; c) \otimes \lf(X \setmin B; c) .
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
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47 |
\]
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
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|
48 |
The first direct sum is indexed by all blobs $B\subset X$, and the second
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
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49 |
by all boundary conditions $c \in \cC(\bd B)$.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
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|
50 |
Note that $\bc_1(X)$ is spanned by 1-blob diagrams $(B, u, r)$.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
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51 |
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52 |
Define the boundary map $\bd : \bc_1(X) \to \bc_0(X)$ by
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
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|
53 |
\[
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
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|
54 |
(B, u, r) \mapsto u\bullet r,
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
55 |
\]
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|
56 |
where $u\bullet r$ denotes the field on $X$ obtained by gluing $u$ to $r$.
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57 |
In other words $\bd : \bc_1(X) \to \bc_0(X)$ is given by
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
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|
58 |
just erasing the blob from the picture
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
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|
59 |
(but keeping the blob label $u$).
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
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|
60 |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
61 |
Note that the skein space $A(X)$
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
62 |
is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
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63 |
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216
|
64 |
$\bc_2(X)$ is, roughly, the space of all relations (redundancies, syzygies) among the
|
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
65 |
local relations encoded in $\bc_1(X)$.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
66 |
More specifically, $\bc_2(X)$ is the space of all finite linear combinations of
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
67 |
2-blob diagrams, of which there are two types, disjoint and nested.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
68 |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
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|
69 |
A disjoint 2-blob diagram consists of
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
70 |
\begin{itemize}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
71 |
\item A pair of closed balls (blobs) $B_0, B_1 \sub X$ with disjoint interiors.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
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|
72 |
\item A field $r \in \cC(X \setmin (B_0 \cup B_1); c_0, c_1)$
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parents:
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|
73 |
(where $c_i \in \cC(\bd B_i)$).
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
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|
74 |
\item Local relation fields $u_i \in U(B_i; c_i)$, $i=1,2$. \nn{We're inconsistent with the indexes -- are they 0,1 or 1,2? I'd prefer 1,2.}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
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changeset
|
75 |
\end{itemize}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
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|
76 |
(See Figure \ref{blob2ddiagram}.)
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|
77 |
\begin{figure}[t]\begin{equation*}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
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|
78 |
\mathfig{.9}{definition/disjoint-blobs}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
79 |
\end{equation*}\caption{A disjoint 2-blob diagram.}\label{blob2ddiagram}\end{figure}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
80 |
We also identify $(B_0, B_1, u_0, u_1, r)$ with $-(B_1, B_0, u_1, u_0, r)$;
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
81 |
reversing the order of the blobs changes the sign.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
82 |
Define $\bd(B_0, B_1, u_0, u_1, r) =
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
83 |
(B_1, u_1, u_0\bullet r) - (B_0, u_0, u_1\bullet r) \in \bc_1(X)$.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
84 |
In other words, the boundary of a disjoint 2-blob diagram
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
85 |
is the sum (with alternating signs)
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
86 |
of the two ways of erasing one of the blobs.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
87 |
It's easy to check that $\bd^2 = 0$.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
88 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
89 |
A nested 2-blob diagram consists of
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
90 |
\begin{itemize}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
91 |
\item A pair of nested balls (blobs) $B_0 \sub B_1 \sub X$.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
92 |
\item A field $r \in \cC(X \setmin B_0; c_0)$
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
93 |
(for some $c_0 \in \cC(\bd B_0)$), which is splittable along $\bd B_1$.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
94 |
\item A local relation field $u_0 \in U(B_0; c_0)$.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
95 |
\end{itemize}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
96 |
(See Figure \ref{blob2ndiagram}.)
|
216
|
97 |
\begin{figure}[t]\begin{equation*}
|
215
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
98 |
\mathfig{.9}{definition/nested-blobs}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
99 |
\end{equation*}\caption{A nested 2-blob diagram.}\label{blob2ndiagram}\end{figure}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
100 |
Let $r = r_1 \bullet r'$, where $r_1 \in \cC(B_1 \setmin B_0; c_0, c_1)$
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
101 |
(for some $c_1 \in \cC(B_1)$) and
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
102 |
$r' \in \cC(X \setmin B_1; c_1)$.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
103 |
Define $\bd(B_0, B_1, u_0, r) = (B_1, u_0\bullet r_1, r') - (B_0, u_0, r)$.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
104 |
Note that the requirement that
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
105 |
local relations are an ideal with respect to gluing guarantees that $u_0\bullet r_1 \in U(B_1)$.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
106 |
As in the disjoint 2-blob case, the boundary of a nested 2-blob is the alternating
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
107 |
sum of the two ways of erasing one of the blobs.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
108 |
If we erase the inner blob, the outer blob inherits the label $u_0\bullet r_1$.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
109 |
It is again easy to check that $\bd^2 = 0$.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
110 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
111 |
As with the 1-blob diagrams, in order to get the linear structure correct it is better to define
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
112 |
(officially)
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
113 |
\begin{eqnarray*}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
114 |
\bc_2(X) & \deq &
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
115 |
\left(
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
116 |
\bigoplus_{B_0, B_1 \text{disjoint}} \bigoplus_{c_0, c_1}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
117 |
U(B_0; c_0) \otimes U(B_1; c_1) \otimes \lf(X\setmin (B_0\cup B_1); c_0, c_1)
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
118 |
\right) \\
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
119 |
&& \bigoplus \left(
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
120 |
\bigoplus_{B_0 \subset B_1} \bigoplus_{c_0}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
121 |
U(B_0; c_0) \otimes \lf(X\setmin B_0; c_0)
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
122 |
\right) .
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
123 |
\end{eqnarray*}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
124 |
The final $\lf(X\setmin B_0; c_0)$ above really means fields splittable along $\bd B_1$,
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
125 |
but we didn't feel like introducing a notation for that.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
126 |
For the disjoint blobs, reversing the ordering of $B_0$ and $B_1$ introduces a minus sign
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
127 |
(rather than a new, linearly independent 2-blob diagram).
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
128 |
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
129 |
Now for the general case.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
130 |
A $k$-blob diagram consists of
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
131 |
\begin{itemize}
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
132 |
\item A collection of blobs $B_i \sub X$, $i = 0, \ldots, k-1$.
|
216
|
133 |
For each $i$ and $j$, we require that either $B_i$ and $B_j$ have disjoint interiors or
|
215
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
134 |
$B_i \sub B_j$ or $B_j \sub B_i$.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
135 |
(The case $B_i = B_j$ is allowed.
|
kevin@6e1638ff-ae45-0410-89bd-df963105f760
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If $B_i \sub B_j$ the boundaries of $B_i$ and $B_j$ are allowed to intersect.)
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If a blob has no other blobs strictly contained in it, we call it a twig blob.
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\item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$.
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(These are implied by the data in the next bullets, so we usually
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suppress them from the notation.)
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$c_i$ and $c_j$ must have identical restrictions to $\bd B_i \cap \bd B_j$
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if the latter space is not empty.
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\item A field $r \in \cC(X \setmin B^t; c^t)$,
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where $B^t$ is the union of all the twig blobs and $c^t \in \cC(\bd B^t)$
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is determined by the $c_i$'s.
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$r$ is required to be splittable along the boundaries of all blobs, twigs or not.
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\item For each twig blob $B_j$ a local relation field $u_j \in U(B_j; c_j)$,
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where $c_j$ is the restriction of $c^t$ to $\bd B_j$.
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If $B_i = B_j$ then $u_i = u_j$.
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\end{itemize}
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(See Figure \ref{blobkdiagram}.)
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152 |
\begin{figure}[t]\begin{equation*}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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\mathfig{.9}{definition/k-blobs}
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\end{equation*}\caption{A $k$-blob diagram.}\label{blobkdiagram}\end{figure}
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155 |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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If two blob diagrams $D_1$ and $D_2$
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differ only by a reordering of the blobs, then we identify
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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$D_1 = \pm D_2$, where the sign is the sign of the permutation relating $D_1$ and $D_2$.
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159 |
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160 |
$\bc_k(X)$ is, roughly, all finite linear combinations of $k$-blob diagrams.
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As before, the official definition is in terms of direct sums
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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of tensor products:
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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163 |
\[
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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164 |
\bc_k(X) \deq \bigoplus_{\overline{B}} \bigoplus_{\overline{c}}
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165 |
\left( \otimes_j U(B_j; c_j)\right) \otimes \lf(X \setmin B^t; c^t) .
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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166 |
\]
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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Here $\overline{B}$ runs over all configurations of blobs, satisfying the conditions above.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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168 |
$\overline{c}$ runs over all boundary conditions, again as described above.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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169 |
$j$ 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 $\overline{B}$.
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170 |
|
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|
171 |
The boundary map
|
|
172 |
\[
|
|
173 |
\bd : \bc_k(X) \to \bc_{k-1}(X)
|
|
174 |
\]
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|
175 |
is defined as follows.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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176 |
Let $b = (\{B_i\}, \{u_j\}, r)$ be a $k$-blob diagram.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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177 |
Let $E_j(b)$ denote the result of erasing the $j$-th blob.
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178 |
If $B_j$ is not a twig blob, this involves only decrementing
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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179 |
the indices of blobs $B_{j+1},\ldots,B_{k-1}$.
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180 |
If $B_j$ is a twig blob, we have to assign new local relation labels
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
181 |
if removing $B_j$ creates new twig blobs.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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182 |
If $B_l$ becomes a twig after removing $B_j$, then set $u_l = u_j\bullet r_l$,
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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183 |
where $r_l$ is the restriction of $r$ to $B_l \setmin B_j$.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
184 |
Finally, define
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
185 |
\eq{
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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186 |
\bd(b) = \sum_{j=0}^{k-1} (-1)^j E_j(b).
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
187 |
}
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
188 |
The $(-1)^j$ factors imply that the terms of $\bd^2(b)$ all cancel.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
189 |
Thus we have a chain complex.
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
190 |
|
216
|
191 |
We note that blob diagrams in $X$ have a structure similar to that of a simplicial set,
|
|
192 |
but with simplices replaced by a more general class of combinatorial shapes.
|
|
193 |
Let $P$ be the minimal set of (isomorphisms classes of) polyhedra which is closed under products
|
|
194 |
and cones, and which contains the point.
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|
195 |
We can associate an element $p(b)$ of $P$ to each blob diagram $b$
|
|
196 |
(equivalently, to each rooted tree) according to the following rules:
|
|
197 |
\begin{itemize}
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|
198 |
\item $p(\emptyset) = pt$, where $\emptyset$ denotes a 0-blob diagram or empty tree;
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|
199 |
\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
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|
200 |
\item $p(\bar{b}) = \kone(p(b))$, where $\bar{b}$ is obtained from $b$ by adding an outer blob which encloses all the others.
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|
201 |
\end{itemize}
|
|
202 |
For example, a diagram of $k$ strictly nested blobs corresponds to a $k$-simplex, while
|
|
203 |
a diagram of $k$ disjoint blobs corresponds to a $k$-cube.
|
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|
204 |
(This correspondence works best if we thing of each twig label $u_i$ as having the form
|
|
205 |
$x - s(e(x))$, where $x$ is an arbitrary field on $B_i$, $e: \cC(B_i) \to C$ is the evaluation map,
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|
206 |
and $s:C \to \cC(B_i)$ is some fixed section of $e$.)
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
207 |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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208 |
|