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\documentclass[11pt,leqno]{article}
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\usepackage{amsmath,amssymb,amsthm}
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\usepackage[all]{xy}
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%%%%% excerpts from my include file of standard macros
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\def\bc{{\cal B}}
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\def\z{\mathbb{Z}}
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\def\r{\mathbb{R}}
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\def\c{\mathbb{C}}
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\def\t{\mathbb{T}}
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\def\du{\sqcup}
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\def\bd{\partial}
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\def\sub{\subset}
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\def\sup{\supset}
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%\def\setmin{\smallsetminus}
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\def\setmin{\setminus}
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\def\ep{\epsilon}
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\def\sgl{_\mathrm{gl}}
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\def\deq{\stackrel{\mathrm{def}}{=}}
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\def\pd#1#2{\frac{\partial #1}{\partial #2}}
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\def\nn#1{{{\it \small [#1]}}}
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% equations
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\newcommand{\eq}[1]{\begin{displaymath}#1\end{displaymath}}
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\newcommand{\eqar}[1]{\begin{eqnarray*}#1\end{eqnarray*}}
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\newcommand{\eqspl}[1]{\begin{displaymath}\begin{split}#1\end{split}\end{displaymath}}
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% tricky way to iterate macros over a list
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\def\semicolon{;}
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\def\applytolist#1{
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\expandafter\def\csname multi#1\endcsname##1{
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\def\multiack{##1}\ifx\multiack\semicolon
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\def\next{\relax}
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\else
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\csname #1\endcsname{##1}
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\def\next{\csname multi#1\endcsname}
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\fi
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\next}
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\csname multi#1\endcsname}
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% \def\cA{{\cal A}} for A..Z
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\def\calc#1{\expandafter\def\csname c#1\endcsname{{\cal #1}}}
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\applytolist{calc}QWERTYUIOPLKJHGFDSAZXCVBNM;
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% \DeclareMathOperator{\pr}{pr} etc.
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\def\declaremathop#1{\expandafter\DeclareMathOperator\csname #1\endcsname{#1}}
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\applytolist{declaremathop}{pr}{im}{id}{gl}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{End}{Hom}{Mat}{Tet}{cat}{Diff}{sign};
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%%%%%% end excerpt
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\title{Blob Homology}
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\begin{document}
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\makeatletter
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\@addtoreset{equation}{section}
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\gdef\theequation{\thesection.\arabic{equation}}
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\makeatother
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\newtheorem{thm}[equation]{Theorem}
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\newtheorem{prop}[equation]{Proposition}
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\newtheorem{lemma}[equation]{Lemma}
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\newtheorem{cor}[equation]{Corollary}
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\newtheorem{defn}[equation]{Definition}
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\maketitle
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\section{Introduction}
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(motivation, summary/outline, etc.)
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(motivation:
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(1) restore exactness in pictures-mod-relations;
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(1') add relations-amongst-relations etc. to pictures-mod-relations;
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(2) want answer independent of handle decomp (i.e. don't
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just go from coend to derived coend (e.g. Hochschild homology));
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(3) ...
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)
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\section{Definitions}
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\subsection{Fields}
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Fix a top dimension $n$.
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A {\it system of fields}
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\nn{maybe should look for better name; but this is the name I use elsewhere}
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is a collection of functors $\cC$ from manifolds of dimension $n$ or less
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to sets.
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These functors must satisfy various properties (see KW TQFT notes for details).
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For example:
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there is a canonical identification $\cC(X \du Y) = \cC(X) \times \cC(Y)$;
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there is a restriction map $\cC(X) \to \cC(\bd X)$;
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gluing manifolds corresponds to fibered products of fields;
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given a field $c \in \cC(Y)$ there is a ``product field"
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$c\times I \in \cC(Y\times I)$; ...
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\nn{should eventually include full details of definition of fields.}
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\nn{note: probably will suppress from notation the distinction
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between fields and their (orientation-reversal) duals}
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\nn{remark that if top dimensional fields are not already linear
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then we will soon linearize them(?)}
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The definition of a system of fields is intended to generalize
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the relevant properties of the following two examples of fields.
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The first example: Fix a target space $B$ and define $\cC(X)$ (where $X$
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is a manifold of dimension $n$ or less) to be the set of
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all maps from $X$ to $B$.
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The second example will take longer to explain.
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Given an $n$-category $C$ with the right sort of duality
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(e.g. pivotal 2-category, 1-category with duals, star 1-category, disklike $n$-category),
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we can construct a system of fields as follows.
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Roughly speaking, $\cC(X)$ will the set of all embedded cell complexes in $X$
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with codimension $i$ cells labeled by $i$-morphisms of $C$.
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We'll spell this out for $n=1,2$ and then describe the general case.
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If $X$ has boundary, we require that the cell decompositions are in general
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position with respect to the boundary --- the boundary intersects each cell
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transversely, so cells meeting the boundary are mere half-cells.
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Put another way, the cell decompositions we consider are dual to standard cell
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decompositions of $X$.
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We will always assume that our $n$-categories have linear $n$-morphisms.
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146 |
For $n=1$, a field on a 0-manifold $P$ is a labeling of each point of $P$ with
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
147 |
an object (0-morphism) of the 1-category $C$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
148 |
A field on a 1-manifold $S$ consists of
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
149 |
\begin{itemize}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
150 |
\item A cell decomposition of $S$ (equivalently, a finite collection
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
151 |
of points in the interior of $S$);
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
152 |
\item a labeling of each 1-cell (and each half 1-cell adjacent to $\bd S$)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
153 |
by an object (0-morphism) of $C$;
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
154 |
\item a transverse orientation of each 0-cell, thought of as a choice of
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
155 |
``domain" and ``range" for the two adjacent 1-cells; and
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
156 |
\item a labeling of each 0-cell by a morphism (1-morphism) of $C$, with
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
157 |
domain and range determined by the transverse orientation and the labelings of the 1-cells.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
158 |
\end{itemize}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
159 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
160 |
If $C$ is an algebra (i.e. if $C$ has only one 0-morphism) we can ignore the labels
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
161 |
of 1-cells, so a field on a 1-manifold $S$ is a finite collection of points in the
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
162 |
interior of $S$, each transversely oriented and each labeled by an element (1-morphism)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
163 |
of the algebra.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
164 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
165 |
For $n=2$, a field on a 0-manifold $P$ is a labeling of each point of $P$ with
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
166 |
an object of the 2-category $C$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
167 |
A field of a 1-manifold is defined as in the $n=1$ case, using the 0- and 1-morphisms of $C$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
168 |
A field on a 2-manifold $Y$ consists of
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
169 |
\begin{itemize}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
170 |
\item A cell decomposition of $Y$ (equivalently, a graph embedded in $Y$ such
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
171 |
that each component of the complement is homeomorphic to a disk);
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
172 |
\item a labeling of each 2-cell (and each half 2-cell adjacent to $\bd Y$)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
173 |
by a 0-morphism of $C$;
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
174 |
\item a transverse orientation of each 1-cell, thought of as a choice of
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
175 |
``domain" and ``range" for the two adjacent 2-cells;
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
176 |
\item a labeling of each 1-cell by a 1-morphism of $C$, with
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
177 |
domain and range determined by the transverse orientation of the 1-cell
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
178 |
and the labelings of the 2-cells;
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
179 |
\item for each 0-cell, a homeomorphism of the boundary $R$ of a small neighborhood
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
180 |
of the 0-cell to $S^1$ such that the intersections of the 1-cells with $R$ are not mapped
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
181 |
to $\pm 1 \in S^1$; and
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
182 |
\item a labeling of each 0-cell by a 2-morphism of $C$, with domain and range
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
183 |
determined by the labelings of the 1-cells and the parameterizations of the previous
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
184 |
bullet.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
185 |
\end{itemize}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
186 |
\nn{need to say this better; don't try to fit everything into the bulleted list}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
187 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
188 |
For general $n$, a field on a $k$-manifold $X^k$ consists of
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
189 |
\begin{itemize}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
190 |
\item A cell decomposition of $X$;
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
191 |
\item an explicit general position homeomorphism from the link of each $j$-cell
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
192 |
to the boundary of the standard $(k-j)$-dimensional bihedron; and
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
193 |
\item a labeling of each $j$-cell by a $(k-j)$-dimensional morphism of $C$, with
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
194 |
domain and range determined by the labelings of the link of $j$-cell.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
195 |
\end{itemize}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
196 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
197 |
\nn{next definition might need some work; I think linearity relations should
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
198 |
be treated differently (segregated) from other local relations, but I'm not sure
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
199 |
the next definition is the best way to do it}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
200 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
201 |
For top dimensional ($n$-dimensional) manifolds, we're actually interested
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
202 |
in the linearized space of fields.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
203 |
By default, define $\cC_l(X) = \c[\cC(X)]$; that is, $\cC_l(X)$ is
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
204 |
the vector space of finite
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
205 |
linear combinations of fields on $X$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
206 |
If $X$ has boundary, we of course fix a boundary condition: $\cC_l(X; a) = \c[\cC(X; a)]$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
207 |
Thus the restriction (to boundary) maps are well defined because we never
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
208 |
take linear combinations of fields with differing boundary conditions.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
209 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
210 |
In some cases we don't linearize the default way; instead we take the
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
211 |
spaces $\cC_l(X; a)$ to be part of the data for the system of fields.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
212 |
In particular, for fields based on linear $n$-category pictures we linearize as follows.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
213 |
Define $\cC_l(X; a) = \c[\cC(X; a)]/K$, where $K$ is the space generated by
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
214 |
obvious relations on 0-cell labels.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
215 |
More specifically, let $L$ be a cell decomposition of $X$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
216 |
and let $p$ be a 0-cell of $L$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
217 |
Let $\alpha_c$ and $\alpha_d$ be two labelings of $L$ which are identical except that
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
218 |
$\alpha_c$ labels $p$ by $c$ and $\alpha_d$ labels $p$ by $d$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
219 |
Then the subspace $K$ is generated by things of the form
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
220 |
$\lambda \alpha_c + \alpha_d - \alpha_{\lambda c + d}$, where we leave it to the reader
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
221 |
to infer the meaning of $\alpha_{\lambda c + d}$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
222 |
Note that we are still assuming that $n$-categories have linear spaces of $n$-morphisms.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
223 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
224 |
\nn{Maybe comment further: if there's a natural basis of morphisms, then no need;
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
225 |
will do something similar below; in general, whenever a label lives in a linear
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
226 |
space we do something like this; ? say something about tensor
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
227 |
product of all the linear label spaces? Yes:}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
228 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
229 |
For top dimensional ($n$-dimensional) manifolds, we linearize as follows.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
230 |
Define an ``almost-field" to be a field without labels on the 0-cells.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
231 |
(Recall that 0-cells are labeled by $n$-morphisms.)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
232 |
To each unlabeled 0-cell in an almost field there corresponds a (linear) $n$-morphism
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
233 |
space determined by the labeling of the link of the 0-cell.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
234 |
(If the 0-cell were labeled, the label would live in this space.)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
235 |
We associate to each almost-labeling the tensor product of these spaces (one for each 0-cell).
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
236 |
We now define $\cC_l(X; a)$ to be the direct sum over all almost labelings of the
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
237 |
above tensor products.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
238 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
239 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
240 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
241 |
\subsection{Local relations}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
242 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
243 |
Let $B^n$ denote the standard $n$-ball.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
244 |
A {\it local relation} is a collection subspaces $U(B^n; c) \sub \cC_l(B^n; c)$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
245 |
(for all $c \in \cC(\bd B^n)$) satisfying the following (three?) properties.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
246 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
247 |
\nn{implies (extended?) isotopy; stable under gluing; open covers?; ...}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
248 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
249 |
For maps into spaces, $U(B^n; c)$ is generated by things of the form $a-b \in \cC_l(B^n; c)$,
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
250 |
where $a$ and $b$ are maps (fields) which are homotopic rel boundary.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
251 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
252 |
For $n$-category pictures, $U(B^n; c)$ is equal to the kernel of the evaluation map
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
253 |
$\cC_l(B^n; c) \to \mor(c', c'')$, where $(c', c'')$ is some (any) division of $c$ into
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
254 |
domain and range.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
255 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
256 |
\nn{maybe examples of local relations before general def?}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
257 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
258 |
Note that the $Y$ is an $n$-manifold which is merely homeomorphic to the standard $B^n$,
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
259 |
then any homeomorphism $B^n \to Y$ induces the same local subspaces for $Y$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
260 |
We'll denote these by $U(Y; c) \sub \cC_l(Y; c)$, $c \in \cC(\bd Y)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
261 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
262 |
Given a system of fields and local relations, we define the skein space
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
263 |
$A(Y^n; c)$ to be the space of all finite linear combinations of fields on
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
264 |
the $n$-manifold $Y$ modulo local relations.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
265 |
The Hilbert space $Z(Y; c)$ for the TQFT based on the fields and local relations
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
266 |
is defined to be the dual of $A(Y; c)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
267 |
(See KW TQFT notes or xxxx for details.)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
268 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
269 |
The blob complex is in some sense the derived version of $A(Y; c)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
270 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
271 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
272 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
273 |
\subsection{The blob complex}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
274 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
275 |
Let $X$ be an $n$-manifold.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
276 |
Assume a fixed system of fields.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
277 |
In this section we will usually suppress boundary conditions on $X$ from the notation
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
278 |
(e.g. write $\cC_l(X)$ instead of $\cC_l(X; c)$).
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
279 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
280 |
We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
281 |
submanifold of $X$, then $X \setmin Y$ implicitly means the closure
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
282 |
$\overline{X \setmin Y}$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
283 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
284 |
We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
285 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
286 |
Define $\bc_0(X) = \cC_l(X)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
287 |
(If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \cC_l(X; c)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
288 |
We'll omit this sort of detail in the rest of this section.)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
289 |
In other words, $\bc_0(X)$ is just the space of all linearized fields on $X$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
290 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
291 |
$\bc_1(X)$ is the space of all local relations that can be imposed on $\bc_0(X)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
292 |
More specifically, define a 1-blob diagram to consist of
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
293 |
\begin{itemize}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
294 |
\item An embedded closed ball (``blob") $B \sub X$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
295 |
%\nn{Does $B$ need a homeo to the standard $B^n$? I don't think so.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
296 |
%(See note in previous subsection.)}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
297 |
%\item A field (boundary condition) $c \in \cC(\bd B) = \cC(\bd(X \setmin B))$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
298 |
\item A field $r \in \cC(X \setmin B; c)$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
299 |
(for some $c \in \cC(\bd B) = \cC(\bd(X \setmin B))$).
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
300 |
\item A local relation field $u \in U(B; c)$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
301 |
(same $c$ as previous bullet).
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
302 |
\end{itemize}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
303 |
%(Note that the the field $c$ is determined (implicitly) as the boundary of $u$ and/or $r$,
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
304 |
%so we will omit $c$ from the notation.)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
305 |
Define $\bc_1(X)$ to be the space of all finite linear combinations of
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
306 |
1-blob diagrams, modulo the simple relations relating labels of 0-cells and
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
307 |
also the label ($u$ above) of the blob.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
308 |
\nn{maybe spell this out in more detail}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
309 |
(See xxxx above.)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
310 |
\nn{maybe restate this in terms of direct sums of tensor products.}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
311 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
312 |
There is a map $\bd : \bc_1(X) \to \bc_0(X)$ which sends $(B, r, u)$ to $ru$, the linear
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
313 |
combination of fields on $X$ obtained by gluing $r$ to $u$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
314 |
In other words $\bd : \bc_1(X) \to \bc_0(X)$ is given by
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
315 |
just erasing the blob from the picture
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
316 |
(but keeping the blob label $u$).
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
317 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
318 |
Note that the skein module $A(X)$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
319 |
is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
320 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
321 |
$\bc_2(X)$ is the space of all relations (redundancies) among the relations of $\bc_1(X)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
322 |
More specifically, $\bc_2(X)$ is the space of all finite linear combinations of
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
323 |
2-blob diagrams (defined below), modulo the usual linear label relations.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
324 |
\nn{and also modulo blob reordering relations?}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
325 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
326 |
\nn{maybe include longer discussion to motivate the two sorts of 2-blob diagrams}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
327 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
328 |
There are two types of 2-blob diagram: disjoint and nested.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
329 |
A disjoint 2-blob diagram consists of
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
330 |
\begin{itemize}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
331 |
\item A pair of disjoint closed balls (blobs) $B_0, B_1 \sub X$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
332 |
%\item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
333 |
\item A field $r \in \cC(X \setmin (B_0 \cup B_1); c_0, c_1)$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
334 |
(where $c_i \in \cC(\bd B_i)$).
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
335 |
\item Local relation fields $u_i \in U(B_i; c_i)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
336 |
\end{itemize}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
337 |
Define $\bd(B_0, B_1, r, u_0, u_1) = (B_1, ru_0, u_1) - (B_0, ru_1, u_0) \in \bc_1(X)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
338 |
In other words, the boundary of a disjoint 2-blob diagram
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
339 |
is the sum (with alternating signs)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
340 |
of the two ways of erasing one of the blobs.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
341 |
It's easy to check that $\bd^2 = 0$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
342 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
343 |
A nested 2-blob diagram consists of
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
344 |
\begin{itemize}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
345 |
\item A pair of nested balls (blobs) $B_0 \sub B_1 \sub X$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
346 |
\item A field $r \in \cC(X \setmin B_0; c_0)$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
347 |
(for some $c_0 \in \cC(\bd B_0)$).
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
348 |
Let $r = r_1 \cup r'$, where $r_1 \in \cC(B_1 \setmin B_0; c_0, c_1)$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
349 |
(for some $c_1 \in \cC(B_1)$) and
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
350 |
$r' \in \cC(X \setmin B_1; c_1)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
351 |
\item A local relation field $u_0 \in U(B_0; c_0)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
352 |
\end{itemize}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
353 |
Define $\bd(B_0, B_1, r, u_0) = (B_1, r', r_1u_0) - (B_0, r, u_0)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
354 |
Note that xxxx above guarantees that $r_1u_0 \in U(B_1)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
355 |
As in the disjoint 2-blob case, the boundary of a nested 2-blob is the alternating
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
356 |
sum of the two ways of erasing one of the blobs.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
357 |
If we erase the inner blob, the outer blob inherits the label $r_1u_0$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
358 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
359 |
Now for the general case.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
360 |
A $k$-blob diagram consists of
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
361 |
\begin{itemize}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
362 |
\item A collection of blobs $B_i \sub X$, $i = 0, \ldots, k-1$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
363 |
For each $i$ and $j$, we require that either $B_i \cap B_j$ is empty or
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
364 |
$B_i \sub B_j$ or $B_j \sub B_i$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
365 |
(The case $B_i = B_j$ is allowed.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
366 |
If $B_i \sub B_j$ the boundaries of $B_i$ and $B_j$ are allowed to intersect.)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
367 |
If a blob has no other blobs strictly contained in it, we call it a twig blob.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
368 |
%\item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
369 |
%(These are implied by the data in the next bullets, so we usually
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
370 |
%suppress them from the notation.)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
371 |
%$c_i$ and $c_j$ must have identical restrictions to $\bd B_i \cap \bd B_j$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
372 |
%if the latter space is not empty.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
373 |
\item A field $r \in \cC(X \setmin B^t; c^t)$,
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
374 |
where $B^t$ is the union of all the twig blobs and $c^t \in \cC(\bd B^t)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
375 |
\item For each twig blob $B_j$ a local relation field $u_j \in U(B_j; c_j)$,
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
376 |
where $c_j$ is the restriction of $c^t$ to $\bd B_j$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
377 |
If $B_i = B_j$ then $u_i = u_j$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
378 |
\end{itemize}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
379 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
380 |
We define $\bc_k(X)$ to be the vector space of all finite linear combinations
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
381 |
of $k$-blob diagrams, modulo the linear label relations and
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
382 |
blob reordering relations defined in the remainder of this paragraph.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
383 |
Let $x$ be a blob diagram with one undetermined $n$-morphism label.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
384 |
The unlabeled entity is either a blob or a 0-cell outside of the twig blobs.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
385 |
Let $a$ and $b$ be two possible $n$-morphism labels for
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
386 |
the unlabeled blob or 0-cell.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
387 |
Let $c = \lambda a + b$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
388 |
Let $x_a$ be the blob diagram with label $a$, and define $x_b$ and $x_c$ similarly.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
389 |
Then we impose the relation
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
390 |
\eq{
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
391 |
x_c = \lambda x_a + x_b .
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
392 |
}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
393 |
\nn{should do this in terms of direct sums of tensor products}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
394 |
Let $x$ and $x'$ be two blob diagrams which differ only by a permutation $\pi$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
395 |
of their blob labelings.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
396 |
Then we impose the relation
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
397 |
\eq{
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
398 |
x = \sign(\pi) x' .
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
399 |
}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
400 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
401 |
(Alert readers will have noticed that for $k=2$ our definition
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
402 |
of $\bc_k(X)$ is slightly different from the previous definition
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
403 |
of $\bc_2(X)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
404 |
The general definition takes precedence;
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
405 |
the earlier definition was simplified for purposes of exposition.)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
406 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
407 |
The boundary map $\bd : \bc_k(X) \to \bc_{k-1}(X)$ is defined as follows.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
408 |
Let $b = (\{B_i\}, r, \{u_j\})$ be a $k$-blob diagram.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
409 |
Let $E_j(b)$ denote the result of erasing the $j$-th blob.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
410 |
If $B_j$ is not a twig blob, this involves only decrementing
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
411 |
the indices of blobs $B_{j+1},\ldots,B_{k-1}$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
412 |
If $B_j$ is a twig blob, we have to assign new local relation labels
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
413 |
if removing $B_j$ creates new twig blobs.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
414 |
If $B_l$ becomes a twig after removing $B_j$, then set $u_l = r_lu_j$,
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
415 |
where $r_l$ is the restriction of $r$ to $B_l \setmin B_j$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
416 |
Finally, define
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
417 |
\eq{
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
418 |
\bd(b) = \sum_{j=0}^{k-1} (-1)^j E_j(b).
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
419 |
}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
420 |
The $(-1)^j$ factors imply that the terms of $\bd^2(b)$ all cancel.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
421 |
Thus we have a chain complex.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
422 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
423 |
\nn{?? say something about the ``shape" of tree? (incl = cone, disj = product)}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
424 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
425 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
426 |
\nn{TO DO: ((?)) allow $n$-morphisms to be chain complex instead of just
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
427 |
a vector space; relations to Chas-Sullivan string stuff}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
428 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
429 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
430 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
431 |
\section{Basic properties of the blob complex}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
432 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
433 |
\begin{prop} \label{disjunion}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
434 |
There is a natural isomorphism $\bc_*(X \du Y) \cong \bc_*(X) \otimes \bc_*(Y)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
435 |
\end{prop}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
436 |
\begin{proof}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
437 |
Given blob diagrams $b_1$ on $X$ and $b_2$ on $Y$, we can combine them
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
438 |
(putting the $b_1$ blobs before the $b_2$ blobs in the ordering) to get a
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
439 |
blob diagram $(b_1, b_2)$ on $X \du Y$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
440 |
Because of the blob reordering relations, all blob diagrams on $X \du Y$ arise this way.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
441 |
In the other direction, any blob diagram on $X\du Y$ is equal (up to sign)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
442 |
to one that puts $X$ blobs before $Y$ blobs in the ordering, and so determines
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
443 |
a pair of blob diagrams on $X$ and $Y$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
444 |
These two maps are compatible with our sign conventions \nn{say more about this?} and
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
445 |
with the linear label relations.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
446 |
The two maps are inverses of each other.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
447 |
\nn{should probably say something about sign conventions for the differential
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
448 |
in a tensor product of chain complexes; ask Scott}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
449 |
\end{proof}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
450 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
451 |
For the next proposition we will temporarily restore $n$-manifold boundary
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
452 |
conditions to the notation.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
453 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
454 |
Suppose that for all $c \in \cC(\bd B^n)$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
455 |
we have a splitting $s: H_0(\bc_*(B^n, c)) \to \bc_0(B^n; c)$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
456 |
of the quotient map
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
457 |
$p: \bc_0(B^n; c) \to H_0(\bc_*(B^n, c))$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
458 |
\nn{always the case if we're working over $\c$}.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
459 |
Then
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
460 |
\begin{prop} \label{bcontract}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
461 |
For all $c \in \cC(\bd B^n)$ the natural map $p: \bc_*(B^n, c) \to H_0(\bc_*(B^n, c))$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
462 |
is a chain homotopy equivalence
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
463 |
with inverse $s: H_0(\bc_*(B^n, c)) \to \bc_*(B^n; c)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
464 |
Here we think of $H_0(\bc_*(B^n, c))$ as a 1-step complex concentrated in degree 0.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
465 |
\end{prop}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
466 |
\begin{proof}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
467 |
By assumption $p\circ s = \id$, so all that remains is to find a degree 1 map
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
468 |
$h : \bc_*(B^n; c) \to \bc_*(B^n; c)$ such that $\bd h + h\bd = \id - s \circ p$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
469 |
For $i \ge 1$, define $h_i : \bc_i(B^n; c) \to \bc_{i+1}(B^n; c)$ by adding
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
470 |
an $(i{+}1)$-st blob equal to all of $B^n$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
471 |
In other words, add a new outermost blob which encloses all of the others.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
472 |
Define $h_0 : \bc_0(B^n; c) \to \bc_1(B^n; c)$ by setting $h_0(x)$ equal to
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
473 |
the 1-blob with blob $B^n$ and label $x - s(p(x)) \in U(B^n; c)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
474 |
\nn{$x$ is a 0-blob diagram, i.e. $x \in \cC(B^n; c)$}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
475 |
\end{proof}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
476 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
477 |
(Note that for the above proof to work, we need the linear label relations
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
478 |
for blob labels.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
479 |
Also we need to blob reordering relations (?).)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
480 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
481 |
(Note also that if there is no splitting $s$, we can let $h_0 = 0$ and get a homotopy
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
482 |
equivalence to the 2-step complex $U(B^n; c) \to \cC(B^n; c)$.)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
483 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
484 |
(For fields based on $n$-cats, $H_0(\bc_*(B^n; c)) \cong \mor(c', c'')$.)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
485 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
486 |
\medskip
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
487 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
488 |
As we noted above,
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
489 |
\begin{prop}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
490 |
There is a natural isomorphism $H_0(\bc_*(X)) \cong A(X)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
491 |
\qed
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
492 |
\end{prop}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
493 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
494 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
495 |
\begin{prop}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
496 |
For fixed fields ($n$-cat), $\bc_*$ is a functor from the category
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
497 |
of $n$-manifolds and diffeomorphisms to the category of chain complexes and
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
498 |
(chain map) isomorphisms.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
499 |
\qed
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
500 |
\end{prop}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
501 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
502 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
503 |
In particular,
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
504 |
\begin{prop} \label{diff0prop}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
505 |
There is an action of $\Diff(X)$ on $\bc_*(X)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
506 |
\qed
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
507 |
\end{prop}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
508 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
509 |
The above will be greatly strengthened in Section \ref{diffsect}.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
510 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
511 |
\medskip
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
512 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
513 |
For the next proposition we will temporarily restore $n$-manifold boundary
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
514 |
conditions to the notation.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
515 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
516 |
Let $X$ be an $n$-manifold, $\bd X = Y \cup (-Y) \cup Z$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
517 |
Gluing the two copies of $Y$ together yields an $n$-manifold $X\sgl$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
518 |
with boundary $Z\sgl$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
519 |
Given compatible fields (pictures, boundary conditions) $a$, $b$ and $c$ on $Y$, $-Y$ and $Z$,
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
520 |
we have the blob complex $\bc_*(X; a, b, c)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
521 |
If $b = -a$ (the orientation reversal of $a$), then we can glue up blob diagrams on
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
522 |
$X$ to get blob diagrams on $X\sgl$:
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
523 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
524 |
\begin{prop}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
525 |
There is a natural chain map
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
526 |
\eq{
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
527 |
\gl: \bigoplus_a \bc_*(X; a, -a, c) \to \bc_*(X\sgl; c\sgl).
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
528 |
}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
529 |
The sum is over all fields $a$ on $Y$ compatible at their
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
530 |
($n{-}2$-dimensional) boundaries with $c$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
531 |
`Natural' means natural with respect to the actions of diffeomorphisms.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
532 |
\qed
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
533 |
\end{prop}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
534 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
535 |
The above map is very far from being an isomorphism, even on homology.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
536 |
This will be fixed in Section \ref{gluesect} below.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
537 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
538 |
An instance of gluing we will encounter frequently below is where $X = X_1 \du X_2$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
539 |
and $X\sgl = X_1 \cup_Y X_2$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
540 |
(Typically one of $X_1$ or $X_2$ is a disjoint union of balls.)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
541 |
For $x_i \in \bc_*(X_i)$, we introduce the notation
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
542 |
\eq{
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
543 |
x_1 \bullet x_2 \deq \gl(x_1 \otimes x_2) .
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
544 |
}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
545 |
Note that we have resumed our habit of omitting boundary labels from the notation.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
546 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
547 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
548 |
\bigskip
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
549 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
550 |
\nn{what else?}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
551 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
552 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
553 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
554 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
555 |
\section{$n=1$ and Hochschild homology}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
556 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
557 |
In this section we analyze the blob complex in dimension $n=1$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
558 |
and find that for $S^1$ the homology of the blob complex is the
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
559 |
Hochschild homology of the category (algebroid) that we started with.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
560 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
561 |
Notation: $HB_i(X) = H_i(\bc_*(X))$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
562 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
563 |
Let us first note that there is no loss of generality in assuming that our system of
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
564 |
fields comes from a category.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
565 |
(Or maybe (???) there {\it is} a loss of generality.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
566 |
Given any system of fields, $A(I; a, b) = \cC(I; a, b)/U(I; a, b)$ can be
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
567 |
thought of as the morphisms of a 1-category $C$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
568 |
More specifically, the objects of $C$ are $\cC(pt)$, the morphisms from $a$ to $b$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
569 |
are $A(I; a, b)$, and composition is given by gluing.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
570 |
If we instead take our fields to be $C$-pictures, the $\cC(pt)$ does not change
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
571 |
and neither does $A(I; a, b) = HB_0(I; a, b)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
572 |
But what about $HB_i(I; a, b)$ for $i > 0$?
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
573 |
Might these higher blob homology groups be different?
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
574 |
Seems unlikely, but I don't feel like trying to prove it at the moment.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
575 |
In any case, we'll concentrate on the case of fields based on 1-category
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
576 |
pictures for the rest of this section.)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
577 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
578 |
(Another question: $\bc_*(I)$ is an $A_\infty$-category.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
579 |
How general of an $A_\infty$-category is it?
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
580 |
Given an arbitrary $A_\infty$-category can one find fields and local relations so
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
581 |
that $\bc_*(I)$ is in some sense equivalent to the original $A_\infty$-category?
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
582 |
Probably not, unless we generalize to the case where $n$-morphisms are complexes.)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
583 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
584 |
Continuing...
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
585 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
586 |
Let $C$ be a *-1-category.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
587 |
Then specializing the definitions from above to the case $n=1$ we have:
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
588 |
\begin{itemize}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
589 |
\item $\cC(pt) = \ob(C)$ .
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
590 |
\item Let $R$ be a 1-manifold and $c \in \cC(\bd R)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
591 |
Then an element of $\cC(R; c)$ is a collection of (transversely oriented)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
592 |
points in the interior
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
593 |
of $R$, each labeled by a morphism of $C$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
594 |
The intervals between the points are labeled by objects of $C$, consistent with
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
595 |
the boundary condition $c$ and the domains and ranges of the point labels.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
596 |
\item There is an evaluation map $e: \cC(I; a, b) \to \mor(a, b)$ given by
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
597 |
composing the morphism labels of the points.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
598 |
\item For $x \in \mor(a, b)$ let $\chi(x) \in \cC(I; a, b)$ be the field with a single
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
599 |
point (at some standard location) labeled by $x$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
600 |
Then the kernel of the evaluation map $U(I; a, b)$ is generated by things of the
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
601 |
form $y - \chi(e(y))$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
602 |
Thus we can, if we choose, restrict the blob twig labels to things of this form.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
603 |
\end{itemize}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
604 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
605 |
We want to show that $HB_*(S^1)$ is naturally isomorphic to the
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
606 |
Hochschild homology of $C$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
607 |
\nn{Or better that the complexes are homotopic
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
608 |
or quasi-isomorphic.}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
609 |
In order to prove this we will need to extend the blob complex to allow points to also
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
610 |
be labeled by elements of $C$-$C$-bimodules.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
611 |
%Given an interval (1-ball) so labeled, there is an evaluation map to some tensor product
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
612 |
%(over $C$) of $C$-$C$-bimodules.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
613 |
%Define the local relations $U(I; a, b)$ to be the direct sum of the kernels of these maps.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
614 |
%Now we can define the blob complex for $S^1$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
615 |
%This complex is the sum of complexes with a fixed cyclic tuple of bimodules present.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
616 |
%If $M$ is a $C$-$C$-bimodule, let $G_*(M)$ denote the summand of $\bc_*(S^1)$ corresponding
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
617 |
%to the cyclic 1-tuple $(M)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
618 |
%In other words, $G_*(M)$ is a blob-like complex where exactly one point is labeled
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
619 |
%by an element of $M$ and the remaining points are labeled by morphisms of $C$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
620 |
%It's clear that $G_*(C)$ is isomorphic to the original bimodule-less
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
621 |
%blob complex for $S^1$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
622 |
%\nn{Is it really so clear? Should say more.}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
623 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
624 |
%\nn{alternative to the above paragraph:}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
625 |
Fix points $p_1, \ldots, p_k \in S^1$ and $C$-$C$-bimodules $M_1, \ldots M_k$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
626 |
We define a blob-like complex $F_*(S^1, (p_i), (M_i))$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
627 |
The fields have elements of $M_i$ labeling $p_i$ and elements of $C$ labeling
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
628 |
other points.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
629 |
The blob twig labels lie in kernels of evaluation maps.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
630 |
(The range of these evaluation maps is a tensor product (over $C$) of $M_i$'s.)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
631 |
Let $F_*(M) = F_*(S^1, (*), (M))$, where $* \in S^1$ is some standard base point.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
632 |
In other words, fields for $F_*(M)$ have an element of $M$ at the fixed point $*$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
633 |
and elements of $C$ at variable other points.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
634 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
635 |
We claim that the homology of $F_*(M)$ is isomorphic to the Hochschild
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
636 |
homology of $M$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
637 |
\nn{Or maybe we should claim that $M \to F_*(M)$ is the/a derived coend.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
638 |
Or maybe that $F_*(M)$ is quasi-isomorphic (or perhaps homotopic) to the Hochschild
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
639 |
complex of $M$.}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
640 |
This follows from the following lemmas:
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
641 |
\begin{itemize}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
642 |
\item $F_*(M_1 \oplus M_2) \cong F_*(M_1) \oplus F_*(M_2)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
643 |
\item An exact sequence $0 \to M_1 \to M_2 \to M_3 \to 0$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
644 |
gives rise to an exact sequence $0 \to F_*(M_1) \to F_*(M_2) \to F_*(M_3) \to 0$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
645 |
(See below for proof.)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
646 |
\item $F_*(C\otimes C)$ (the free $C$-$C$-bimodule with one generator) is
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
647 |
homotopic to the 0-step complex $C$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
648 |
(See below for proof.)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
649 |
\item $F_*(C)$ (here $C$ is wearing its $C$-$C$-bimodule hat) is homotopic to $\bc_*(S^1)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
650 |
(See below for proof.)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
651 |
\end{itemize}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
652 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
653 |
First we show that $F_*(C\otimes C)$ is
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
654 |
homotopic to the 0-step complex $C$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
655 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
656 |
Let $F'_* \sub F_*(C\otimes C)$ be the subcomplex where the label of
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
657 |
the point $*$ is $1 \otimes 1 \in C\otimes C$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
658 |
We will show that the inclusion $i: F'_* \to F_*(C\otimes C)$ is a quasi-isomorphism.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
659 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
660 |
Fix a small $\ep > 0$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
661 |
Let $B_\ep$ be the ball of radius $\ep$ around $* \in S^1$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
662 |
Let $F^\ep_* \sub F_*(C\otimes C)$ be the subcomplex where $B_\ep$ is either disjoint from
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
663 |
or contained in all blobs, and the two boundary points of $B_\ep$ are not labeled points.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
664 |
For a field (picture) $y$ on $B_\ep$, let $s_\ep(y)$ be the equivalent picture with~$*$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
665 |
labeled by $1\otimes 1$ and the only other labeled points at distance $\pm\ep/2$ from $*$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
666 |
(See Figure xxxx.)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
667 |
\nn{maybe it's simpler to assume that there are no labeled points, other than $*$, in $B_\ep$.}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
668 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
669 |
Define a degree 1 chain map $j_\ep : F^\ep_* \to F^\ep_*$ as follows.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
670 |
Let $x \in F^\ep_*$ be a blob diagram.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
671 |
If $*$ is not contained in any twig blob, $j_\ep(x)$ is obtained by adding $B_\ep$ to
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
672 |
$x$ as a new twig blob, with label $y - s_\ep(y)$, where $y$ is the restriction of $x$ to $B_\ep$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
673 |
If $*$ is contained in a twig blob $B$ with label $u = \sum z_i$, $j_\ep(x)$ is obtained as follows.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
674 |
Let $y_i$ be the restriction of $z_i$ to $*$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
675 |
Let $x_i$ be equal to $x$ outside of $B$, equal to $z_i$ on $B \setmin B_\ep$,
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
676 |
and have an additional blob $B_\ep$ with label $y_i - s_\ep(y_i)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
677 |
Define $j_\ep(x) = \sum x_i$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
678 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
679 |
Note that if $x \in F'_* \cap F^\ep_*$ then $j_\ep(x) \in F'_*$ also.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
680 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
681 |
The key property of $j_\ep$ is
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
682 |
\eq{
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
683 |
\bd j_\ep + j_\ep \bd = \id - \sigma_\ep ,
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
684 |
}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
685 |
where $\sigma_\ep : F^\ep_* \to F^\ep_*$ is given by replacing the restriction of each field
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
686 |
mentioned in $x \in F^\ep_*$ (call the restriction $y$) with $s_\ep(y)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
687 |
Note that $\sigma_\ep(x) \in F'$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
688 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
689 |
If $j_\ep$ were defined on all of $F_*(C\otimes C)$, it would show that $\sigma_\ep$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
690 |
is a homotopy inverse to the inclusion $F'_* \to F_*(C\otimes C)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
691 |
One strategy would be to try to stitch together various $j_\ep$ for progressively smaller
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
692 |
$\ep$ and show that $F'_*$ is homotopy equivalent to $F_*(C\otimes C)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
693 |
Instead, we'll be less ambitious and just show that
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
694 |
$F'_*$ is quasi-isomorphic to $F_*(C\otimes C)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
695 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
696 |
If $x$ is a cycle in $F_*(C\otimes C)$, then for sufficiently small $\ep$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
697 |
$x \in F_*^\ep$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
698 |
(This is true for any chain in $F_*(C\otimes C)$, since chains are sums of
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
699 |
finitely many blob diagrams.)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
700 |
Then $x$ is homologous to $s_\ep(x)$, which is in $F'_*$, so the inclusion map
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
701 |
is surjective on homology.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
702 |
If $y \in F_*(C\otimes C)$ and $\bd y = x \in F'_*$, then $y \in F^\ep_*$ for some $\ep$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
703 |
and
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
704 |
\eq{
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
705 |
\bd x = \bd (\sigma_\ep(y) + j_\ep(x)) .
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
706 |
}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
707 |
Since $\sigma_\ep(y) + j_\ep(x) \in F'$, it follows that the inclusion map is injective on homology.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
708 |
This completes the proof that $F'_*$ is quasi-isomorphic to $F_*(C\otimes C)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
709 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
710 |
\medskip
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
711 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
712 |
Let $F''_* \sub F'_*$ be the subcomplex of $F'_*$ where $*$ is not contained in any blob.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
713 |
We will show that the inclusion $i: F''_* \to F'_*$ is a homotopy equivalence.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
714 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
715 |
First, a lemma: Let $G''_*$ and $G'_*$ be defined the same as $F''_*$ and $F'_*$, except with
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
716 |
$S^1$ replaced some (any) neighborhood of $* \in S^1$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
717 |
Then $G''_*$ and $G'_*$ are both contractible.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
718 |
For $G'_*$ the proof is the same as in (\ref{bcontract}), except that the splitting
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
719 |
$G'_0 \to H_0(G'_*)$ concentrates the point labels at two points to the right and left of $*$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
720 |
For $G''_*$ we note that any cycle is supported \nn{need to establish terminology for this; maybe
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
721 |
in ``basic properties" section above} away from $*$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
722 |
Thus any cycle lies in the image of the normal blob complex of a disjoint union
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
723 |
of two intervals, which is contractible by (\ref{bcontract}) and (\ref{disjunion}).
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
724 |
Actually, we need the further (easy) result that the inclusion
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
725 |
$G''_* \to G'_*$ induces an isomorphism on $H_0$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
726 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
727 |
Next we construct a degree 1 map (homotopy) $h: F'_* \to F'_*$ such that
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
728 |
for all $x \in F'_*$ we have
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
729 |
\eq{
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
730 |
x - \bd h(x) - h(\bd x) \in F''_* .
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
731 |
}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
732 |
Since $F'_0 = F''_0$, we can take $h_0 = 0$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
733 |
Let $x \in F'_1$, with single blob $B \sub S^1$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
734 |
If $* \notin B$, then $x \in F''_1$ and we define $h_1(x) = 0$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
735 |
If $* \in B$, then we work in the image of $G'_*$ and $G''_*$ (with respect to $B$).
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
736 |
Choose $x'' \in G''_1$ such that $\bd x'' = \bd x$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
737 |
Since $G'_*$ is contractible, there exists $y \in G'_2$ such that $\bd y = x - x''$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
738 |
Define $h_1(x) = y$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
739 |
The general case is similar, except that we have to take lower order homotopies into account.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
740 |
Let $x \in F'_k$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
741 |
If $*$ is not contained in any of the blobs of $x$, then define $h_k(x) = 0$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
742 |
Otherwise, let $B$ be the outermost blob of $x$ containing $*$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
743 |
By xxxx above, $x = x' \bullet p$, where $x'$ is supported on $B$ and $p$ is supported away from $B$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
744 |
So $x' \in G'_l$ for some $l \le k$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
745 |
Choose $x'' \in G''_l$ such that $\bd x'' = \bd (x' - h_{l-1}\bd x')$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
746 |
Choose $y \in G'_{l+1}$ such that $\bd y = x' - x'' - h_{l-1}\bd x'$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
747 |
Define $h_k(x) = y \bullet p$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
748 |
This completes the proof that $i: F''_* \to F'_*$ is a homotopy equivalence.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
749 |
\nn{need to say above more clearly and settle on notation/terminology}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
750 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
751 |
Finally, we show that $F''_*$ is contractible.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
752 |
\nn{need to also show that $H_0$ is the right thing; easy, but I won't do it now}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
753 |
Let $x$ be a cycle in $F''_*$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
754 |
The union of the supports of the diagrams in $x$ does not contain $*$, so there exists a
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
755 |
ball $B \subset S^1$ containing the union of the supports and not containing $*$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
756 |
Adding $B$ as a blob to $x$ gives a contraction.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
757 |
\nn{need to say something else in degree zero}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
758 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
759 |
This completes the proof that $F_*(C\otimes C)$ is
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
760 |
homotopic to the 0-step complex $C$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
761 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
762 |
\medskip
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
763 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
764 |
Next we show that $F_*(C)$ is homotopic \nn{q-isom?} to $\bc_*(S^1)$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
765 |
\nn{...}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
766 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
767 |
\bigskip
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
768 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
769 |
\nn{still need to prove exactness claim}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
770 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
771 |
\nn{What else needs to be said to establish quasi-isomorphism to Hochschild complex?
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
772 |
Do we need a map from hoch to blob?
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
773 |
Does the above exactness and contractibility guarantee such a map without writing it
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
774 |
down explicitly?
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
775 |
Probably it's worth writing down an explicit map even if we don't need to.}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
776 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
777 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
778 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
779 |
\section{Action of $C_*(\Diff(X))$} \label{diffsect}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
780 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
781 |
Let $CD_*(X)$ denote $C_*(\Diff(X))$, the singular chain complex of
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
782 |
the space of diffeomorphisms
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
783 |
of the $n$-manifold $X$ (fixed on $\bd X$).
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
784 |
For convenience, we will permit the singular cells generating $CD_*(X)$ to be more general
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
785 |
than simplices --- they can be based on any linear polyhedron.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
786 |
\nn{be more restrictive here? does more need to be said?}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
787 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
788 |
\begin{prop} \label{CDprop}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
789 |
For each $n$-manifold $X$ there is a chain map
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
790 |
\eq{
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
791 |
e_X : CD_*(X) \otimes \bc_*(X) \to \bc_*(X) .
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
792 |
}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
793 |
On $CD_0(X) \otimes \bc_*(X)$ it agrees with the obvious action of $\Diff(X)$ on $\bc_*(X)$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
794 |
(Proposition (\ref{diff0prop})).
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
795 |
For any splitting $X = X_1 \cup X_2$, the following diagram commutes
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
796 |
\eq{ \xymatrix{
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
797 |
CD_*(X) \otimes \bc_*(X) \ar[r]^{e_X} & \bc_*(X) \\
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
798 |
CD_*(X_1) \otimes CD_*(X_2) \otimes \bc_*(X_1) \otimes \bc_*(X_2)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
799 |
\ar@/_4ex/[r]_{e_{X_1} \otimes e_{X_2}} \ar[u]^{\gl \otimes \gl} &
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
800 |
\bc_*(X_1) \otimes \bc_*(X_2) \ar[u]_{\gl}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
801 |
} }
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
802 |
Any other map satisfying the above two properties is homotopic to $e_X$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
803 |
\end{prop}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
804 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
805 |
The proof will occupy the remainder of this section.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
806 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
807 |
\medskip
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
808 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
809 |
Let $f: P \times X \to X$ be a family of diffeomorphisms and $S \sub X$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
810 |
We say that {\it $f$ is supported on $S$} if $f(p, x) = f(q, x)$ for all
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
811 |
$x \notin S$ and $p, q \in P$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
812 |
Note that if $f$ is supported on $S$ then it is also supported on any $R \sup S$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
813 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
814 |
Let $\cU = \{U_\alpha\}$ be an open cover of $X$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
815 |
A $k$-parameter family of diffeomorphisms $f: P \times X \to X$ is
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
816 |
{\it adapted to $\cU$} if there is a factorization
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
817 |
\eq{
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
818 |
P = P_1 \times \cdots \times P_m
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
819 |
}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
820 |
(for some $m \le k$)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
821 |
and families of diffeomorphisms
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
822 |
\eq{
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
823 |
f_i : P_i \times X \to X
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
824 |
}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
825 |
such that
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
826 |
\begin{itemize}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
827 |
\item each $f_i(p, \cdot): X \to X$ is supported on some connected $V_i \sub X$;
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
828 |
\item the $V_i$'s are mutually disjoint;
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
829 |
\item each $V_i$ is the union of at most $k_i$ of the $U_\alpha$'s,
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
830 |
where $k_i = \dim(P_i)$; and
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
831 |
\item $f(p, \cdot) = f_1(p_1, \cdot) \circ \cdots \circ f_m(p_m, \cdot)$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
832 |
for all $p = (p_1, \ldots, p_m)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
833 |
\end{itemize}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
834 |
A chain $x \in C_k(\Diff(M))$ is (by definition) adapted to $\cU$ if is is the sum
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
835 |
of singular cells, each of which is adapted to $\cU$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
836 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
837 |
\begin{lemma} \label{extension_lemma}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
838 |
Let $x \in CD_k(X)$ be a singular chain such that $\bd x$ is adapted to $\cU$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
839 |
Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
840 |
\end{lemma}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
841 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
842 |
The proof will be given in Section \ref{fam_diff_sect}.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
843 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
844 |
\medskip
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
845 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
846 |
Let $B_1, \ldots, B_m$ be a collection of disjoint balls in $X$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
847 |
(e.g.~the support of a blob diagram).
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
848 |
We say that $f:P\times X\to X$ is {\it compatible} with $\{B_j\}$ if
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
849 |
$f$ has support a disjoint collection of balls $D_i \sub X$ and for all $i$ and $j$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
850 |
either $B_j \sub D_i$ or $B_j \cap D_i = \emptyset$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
851 |
A chain $x \in CD_k(X)$ is compatible with $\{B_j\}$ if it is a sum of singular cells,
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
852 |
each of which is compatible.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
853 |
(Note that we could strengthen the definition of compatibility to incorporate
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
854 |
a factorization condition, similar to the definition of ``adapted to" above.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
855 |
The weaker definition given here will suffice for our needs below.)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
856 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
857 |
\begin{cor} \label{extension_lemma_2}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
858 |
Let $x \in CD_k(X)$ be a singular chain such that $\bd x$ is compatible with $\{B_j\}$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
859 |
Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is compatible with $\{B_j\}$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
860 |
\end{cor}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
861 |
\begin{proof}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
862 |
This will follow from Lemma \ref{extension_lemma} for
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
863 |
appropriate choice of cover $\cU = \{U_\alpha\}$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
864 |
Let $U_{\alpha_1}, \ldots, U_{\alpha_k}$ be any $k$ open sets of $\cU$, and let
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
865 |
$V_1, \ldots, V_m$ be the connected components of $U_{\alpha_1}\cup\cdots\cup U_{\alpha_k}$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
866 |
Choose $\cU$ fine enough so that there exist disjoint balls $B'_j \sup B_j$ such that for all $i$ and $j$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
867 |
either $V_i \sub B'_j$ or $V_i \cap B'_j = \emptyset$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
868 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
869 |
Apply Lemma \ref{extension_lemma} first to each singular cell $f_i$ of $\bd x$,
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
870 |
with the (compatible) support of $f_i$ in place of $X$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
871 |
This insures that the resulting homotopy $h_i$ is compatible.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
872 |
Now apply Lemma \ref{extension_lemma} to $x + \sum h_i$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
873 |
\nn{actually, need to start with the 0-skeleton of $\bd x$, then 1-skeleton, etc.; fix this}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
874 |
\end{proof}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
875 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
876 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
877 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
878 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
879 |
\section{Families of Diffeomorphisms} \label{fam_diff_sect}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
880 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
881 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
882 |
Lo, the proof of Lemma (\ref{extension_lemma}):
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
883 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
884 |
\nn{should this be an appendix instead?}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
885 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
886 |
\nn{for pedagogical reasons, should do $k=1,2$ cases first; probably do this in
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
887 |
later draft}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
888 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
889 |
\nn{not sure what the best way to deal with boundary is; for now just give main argument, worry
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
890 |
about boundary later}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
891 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
892 |
Recall that we are given
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
893 |
an open cover $\cU = \{U_\alpha\}$ and an
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
894 |
$x \in CD_k(X)$ such that $\bd x$ is adapted to $\cU$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
895 |
We must find a homotopy of $x$ (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
896 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
897 |
Let $\{r_\alpha : X \to [0,1]\}$ be a partition of unity for $\cU$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
898 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
899 |
As a first approximation to the argument we will eventually make, let's replace $x$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
900 |
with a single singular cell
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
901 |
\eq{
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
902 |
f: P \times X \to X .
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
903 |
}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
904 |
Also, we'll ignore for now issues around $\bd P$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
905 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
906 |
Our homotopy will have the form
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
907 |
\eqar{
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
908 |
F: I \times P \times X &\to& X \\
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
909 |
(t, p, x) &\mapsto& f(u(t, p, x), x)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
910 |
}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
911 |
for some function
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
912 |
\eq{
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
913 |
u : I \times P \times X \to P .
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
914 |
}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
915 |
First we describe $u$, then we argue that it does what we want it to do.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
916 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
917 |
For each cover index $\alpha$ choose a cell decomposition $K_\alpha$ of $P$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
918 |
The various $K_\alpha$ should be in general position with respect to each other.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
919 |
We will see below that the $K_\alpha$'s need to be sufficiently fine in order
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
920 |
to insure that $F$ above is a homotopy through diffeomorphisms of $X$ and not
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
921 |
merely a homotopy through maps $X\to X$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
922 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
923 |
Let $L$ be the union of all the $K_\alpha$'s.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
924 |
$L$ is itself a cell decomposition of $P$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
925 |
\nn{next two sentences not needed?}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
926 |
To each cell $a$ of $L$ we associate the tuple $(c_\alpha)$,
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
927 |
where $c_\alpha$ is the codimension of the cell of $K_\alpha$ which contains $c$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
928 |
Since the $K_\alpha$'s are in general position, we have $\sum c_\alpha \le k$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
929 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
930 |
Let $J$ denote the handle decomposition of $P$ corresponding to $L$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
931 |
Each $i$-handle $C$ of $J$ has an $i$-dimensional tangential coordinate and,
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
932 |
more importantly, a $k{-}i$-dimensional normal coordinate.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
933 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
934 |
For each $k$-cell $c$ of each $K_\alpha$, choose a point $p_c \in c \sub P$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
935 |
Let $D$ be a $k$-handle of $J$, and let $d$ also denote the corresponding
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
936 |
$k$-cell of $L$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
937 |
To $D$ we associate the tuple $(c_\alpha)$ of $k$-cells of the $K_\alpha$'s
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
938 |
which contain $d$, and also the corresponding tuple $(p_{c_\alpha})$ of points in $P$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
939 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
940 |
For $p \in D$ we define
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
941 |
\eq{
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
942 |
u(t, p, x) = (1-t)p + t \sum_\alpha r_\alpha(x) p_{c_\alpha} .
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
943 |
}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
944 |
(Recall that $P$ is a single linear cell, so the weighted average of points of $P$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
945 |
makes sense.)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
946 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
947 |
So far we have defined $u(t, p, x)$ when $p$ lies in a $k$-handle of $J$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
948 |
For handles of $J$ of index less than $k$, we will define $u$ to
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
949 |
interpolate between the values on $k$-handles defined above.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
950 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
951 |
If $p$ lies in a $k{-}1$-handle $E$, let $\eta : E \to [0,1]$ be the normal coordinate
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
952 |
of $E$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
953 |
In particular, $\eta$ is equal to 0 or 1 only at the intersection of $E$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
954 |
with a $k$-handle.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
955 |
Let $\beta$ be the index of the $K_\beta$ containing the $k{-}1$-cell
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
956 |
corresponding to $E$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
957 |
Let $q_0, q_1 \in P$ be the points associated to the two $k$-cells of $K_\beta$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
958 |
adjacent to the $k{-}1$-cell corresponding to $E$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
959 |
For $p \in E$, define
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
960 |
\eq{
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
961 |
u(t, p, x) = (1-t)p + t \left( \sum_{\alpha \ne \beta} r_\alpha(x) p_{c_\alpha}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
962 |
+ r_\beta(x) (\eta(p) q_1 + (1-\eta(p)) q_0) \right) .
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
963 |
}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
964 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
965 |
In general, for $E$ a $k{-}j$-handle, there is a normal coordinate
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
966 |
$\eta: E \to R$, where $R$ is some $j$-dimensional polyhedron.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
967 |
The vertices of $R$ are associated to $k$-cells of the $K_\alpha$, and thence to points of $P$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
968 |
If we triangulate $R$ (without introducing new vertices), we can linearly extend
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
969 |
a map from the the vertices of $R$ into $P$ to a map of all of $R$ into $P$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
970 |
Let $\cN$ be the set of all $\beta$ for which $K_\beta$ has a $k$-cell whose boundary meets
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
971 |
the $k{-}j$-cell corresponding to $E$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
972 |
For each $\beta \in \cN$, let $\{q_{\beta i}\}$ be the set of points in $P$ associated to the aforementioned $k$-cells.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
973 |
Now define, for $p \in E$,
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
974 |
\eq{
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
975 |
u(t, p, x) = (1-t)p + t \left(
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
976 |
\sum_{\alpha \notin \cN} r_\alpha(x) p_{c_\alpha}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
977 |
+ \sum_{\beta \in \cN} r_\beta(x) \left( \sum_i \eta_{\beta i}(p) \cdot q_{\beta i} \right)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
978 |
\right) .
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
979 |
}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
980 |
Here $\eta_{\beta i}(p)$ is the weight given to $q_{\beta i}$ by the linear extension
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
981 |
mentioned above.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
982 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
983 |
This completes the definition of $u: I \times P \times X \to P$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
984 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
985 |
\medskip
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
986 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
987 |
Next we verify that $u$ has the desired properties.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
988 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
989 |
Since $u(0, p, x) = p$ for all $p\in P$ and $x\in X$, $F(0, p, x) = f(p, x)$ for all $p$ and $x$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
990 |
Therefore $F$ is a homotopy from $f$ to something.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
991 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
992 |
Next we show that the the $K_\alpha$'s are sufficiently fine cell decompositions,
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
993 |
then $F$ is a homotopy through diffeomorphisms.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
994 |
We must show that the derivative $\pd{F}{x}(t, p, x)$ is non-singular for all $(t, p, x)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
995 |
We have
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
996 |
\eq{
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
997 |
% \pd{F}{x}(t, p, x) = \pd{f}{x}(u(t, p, x), x) + \pd{f}{p}(u(t, p, x), x) \pd{u}{x}(t, p, x) .
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
998 |
\pd{F}{x} = \pd{f}{x} + \pd{f}{p} \pd{u}{x} .
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
999 |
}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1000 |
Since $f$ is a family of diffeomorphisms, $\pd{f}{x}$ is non-singular and
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1001 |
\nn{bounded away from zero, or something like that}.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1002 |
(Recall that $X$ and $P$ are compact.)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1003 |
Also, $\pd{f}{p}$ is bounded.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1004 |
So if we can insure that $\pd{u}{x}$ is sufficiently small, we are done.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1005 |
It follows from Equation xxxx above that $\pd{u}{x}$ depends on $\pd{r_\alpha}{x}$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1006 |
and the differences amongst the various $p_{c_\alpha}$'s and $q_{\beta i}$'s.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1007 |
These differences are small if the cell decompositions $K_\alpha$ are sufficiently fine.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1008 |
This completes the proof that $F$ is a homotopy through diffeomorphisms.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1009 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1010 |
\medskip
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1011 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1012 |
Next we show that for each handle $D \sub P$, $F(1, \cdot, \cdot) : D\times X \to X$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1013 |
is a singular cell adapted to $\cU$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1014 |
This will complete the proof of the lemma.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1015 |
\nn{except for boundary issues and the `$P$ is a cell' assumption}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1016 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1017 |
Let $j$ be the codimension of $D$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1018 |
(Or rather, the codimension of its corresponding cell. From now on we will not make a distinction
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1019 |
between handle and corresponding cell.)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1020 |
Then $j = j_1 + \cdots + j_m$, $0 \le m \le k$,
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1021 |
where the $j_i$'s are the codimensions of the $K_\alpha$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1022 |
cells of codimension greater than 0 which intersect to form $D$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1023 |
We will show that
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1024 |
if the relevant $U_\alpha$'s are disjoint, then
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1025 |
$F(1, \cdot, \cdot) : D\times X \to X$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1026 |
is a product of singular cells of dimensions $j_1, \ldots, j_m$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1027 |
If some of the relevant $U_\alpha$'s intersect, then we will get a product of singular
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1028 |
cells whose dimensions correspond to a partition of the $j_i$'s.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1029 |
We will consider some simple special cases first, then do the general case.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1030 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1031 |
First consider the case $j=0$ (and $m=0$).
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1032 |
A quick look at Equation xxxx above shows that $u(1, p, x)$, and hence $F(1, p, x)$,
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1033 |
is independent of $p \in P$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1034 |
So the corresponding map $D \to \Diff(X)$ is constant.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1035 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1036 |
Next consider the case $j = 1$ (and $m=1$, $j_1=1$).
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1037 |
Now Equation yyyy applies.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1038 |
We can write $D = D'\times I$, where the normal coordinate $\eta$ is constant on $D'$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1039 |
It follows that the singular cell $D \to \Diff(X)$ can be written as a product
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1040 |
of a constant map $D' \to \Diff(X)$ and a singular 1-cell $I \to \Diff(X)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1041 |
The singular 1-cell is supported on $U_\beta$, since $r_\beta = 0$ outside of this set.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1042 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1043 |
Next case: $j=2$, $m=1$, $j_1 = 2$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1044 |
This is similar to the previous case, except that the normal bundle is 2-dimensional instead of
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1045 |
1-dimensional.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1046 |
We have that $D \to \Diff(X)$ is a product of a constant singular $k{-}2$-cell
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1047 |
and a 2-cell with support $U_\beta$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1048 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1049 |
Next case: $j=2$, $m=2$, $j_1 = j_2 = 2$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1050 |
In this case the codimension 2 cell $D$ is the intersection of two
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1051 |
codimension 1 cells, from $K_\beta$ and $K_\gamma$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1052 |
We can write $D = D' \times I \times I$, where the normal coordinates are constant
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1053 |
on $D'$, and the two $I$ factors correspond to $\beta$ and $\gamma$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1054 |
If $U_\beta$ and $U_\gamma$ are disjoint, then we can factor $D$ into a constant $k{-}2$-cell and
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1055 |
two 1-cells, supported on $U_\beta$ and $U_\gamma$ respectively.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1056 |
If $U_\beta$ and $U_\gamma$ intersect, then we can factor $D$ into a constant $k{-}2$-cell and
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1057 |
a 2-cell supported on $U_\beta \cup U_\gamma$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1058 |
\nn{need to check that this is true}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1059 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1060 |
\nn{finally, general case...}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1061 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1062 |
\nn{this completes proof}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1063 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1064 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1065 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1066 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1067 |
\section{$A_\infty$ action on the boundary}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1068 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1069 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1070 |
\section{Gluing} \label{gluesect}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1071 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1072 |
\section{Extension to ...}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1073 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1074 |
(Need to let the input $n$-category $C$ be a graded thing
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1075 |
(e.g.~DGA or $A_\infty$ $n$-category).)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1076 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1077 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1078 |
\section{What else?...}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1079 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1080 |
\begin{itemize}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1081 |
\item Derive Hochschild standard results from blob point of view?
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1082 |
\item $n=2$ examples
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1083 |
\item Kh
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1084 |
\item dimension $n+1$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1085 |
\item should be clear about PL vs Diff; probably PL is better
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1086 |
(or maybe not)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1087 |
\item say what we mean by $n$-category, $A_\infty$ or $E_\infty$ $n$-category
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1088 |
\item something about higher derived coend things (derived 2-coend, e.g.)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1089 |
\end{itemize}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1090 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1091 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1092 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1093 |
\end{document}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1094 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1095 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1096 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1097 |
%Recall that for $n$-category picture fields there is an evaluation map
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1098 |
%$m: \bc_0(B^n; c, c') \to \mor(c, c')$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1099 |
%If we regard $\mor(c, c')$ as a complex concentrated in degree 0, then this becomes a chain
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1100 |
%map $m: \bc_*(B^n; c, c') \to \mor(c, c')$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1101 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1102 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
1103 |
|