--- a/blob1.tex Tue Apr 22 05:13:02 2008 +0000
+++ b/blob1.tex Thu Apr 24 02:56:34 2008 +0000
@@ -1,14 +1,14 @@
-\documentclass[11pt,leqno]{article}
+\documentclass[11pt,leqno]{amsart}
-\usepackage{amsmath,amssymb,amsthm}
-
-\usepackage[all]{xy}
+\newcommand{\pathtotrunk}{./}
+\input{text/article_preamble.tex}
+\input{text/top_matter.tex}
% test edit #3
%%%%% excerpts from my include file of standard macros
-\def\bc{{\cal B}}
+\def\bc{{\mathcal B}}
\def\z{\mathbb{Z}}
\def\r{\mathbb{R}}
@@ -38,23 +38,23 @@
% tricky way to iterate macros over a list
\def\semicolon{;}
\def\applytolist#1{
- \expandafter\def\csname multi#1\endcsname##1{
- \def\multiack{##1}\ifx\multiack\semicolon
- \def\next{\relax}
- \else
- \csname #1\endcsname{##1}
- \def\next{\csname multi#1\endcsname}
- \fi
- \next}
- \csname multi#1\endcsname}
+ \expandafter\def\csname multi#1\endcsname##1{
+ \def\multiack{##1}\ifx\multiack\semicolon
+ \def\next{\relax}
+ \else
+ \csname #1\endcsname{##1}
+ \def\next{\csname multi#1\endcsname}
+ \fi
+ \next}
+ \csname multi#1\endcsname}
% \def\cA{{\cal A}} for A..Z
-\def\calc#1{\expandafter\def\csname c#1\endcsname{{\cal #1}}}
+\def\calc#1{\expandafter\def\csname c#1\endcsname{{\mathcal #1}}}
\applytolist{calc}QWERTYUIOPLKJHGFDSAZXCVBNM;
% \DeclareMathOperator{\pr}{pr} etc.
\def\declaremathop#1{\expandafter\DeclareMathOperator\csname #1\endcsname{#1}}
-\applytolist{declaremathop}{pr}{im}{id}{gl}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{End}{Hom}{Mat}{Tet}{cat}{Diff}{sign};
+\applytolist{declaremathop}{pr}{im}{id}{gl}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{Tet}{cat}{Diff}{sign};
@@ -74,12 +74,6 @@
\@addtoreset{equation}{section}
\gdef\theequation{\thesection.\arabic{equation}}
\makeatother
-\newtheorem{thm}[equation]{Theorem}
-\newtheorem{prop}[equation]{Proposition}
-\newtheorem{lemma}[equation]{Lemma}
-\newtheorem{cor}[equation]{Corollary}
-\newtheorem{defn}[equation]{Definition}
-
\maketitle
@@ -88,10 +82,10 @@
(motivation, summary/outline, etc.)
-(motivation:
+(motivation:
(1) restore exactness in pictures-mod-relations;
(1') add relations-amongst-relations etc. to pictures-mod-relations;
-(2) want answer independent of handle decomp (i.e. don't
+(2) want answer independent of handle decomp (i.e. don't
just go from coend to derived coend (e.g. Hochschild homology));
(3) ...
)
@@ -102,35 +96,35 @@
Fix a top dimension $n$.
-A {\it system of fields}
+A {\it system of fields}
\nn{maybe should look for better name; but this is the name I use elsewhere}
is a collection of functors $\cC$ from manifolds of dimension $n$ or less
to sets.
These functors must satisfy various properties (see KW TQFT notes for details).
-For example:
+For example:
there is a canonical identification $\cC(X \du Y) = \cC(X) \times \cC(Y)$;
there is a restriction map $\cC(X) \to \cC(\bd X)$;
gluing manifolds corresponds to fibered products of fields;
-given a field $c \in \cC(Y)$ there is a ``product field"
+given a field $c \in \cC(Y)$ there is a ``product field"
$c\times I \in \cC(Y\times I)$; ...
\nn{should eventually include full details of definition of fields.}
-\nn{note: probably will suppress from notation the distinction
+\nn{note: probably will suppress from notation the distinction
between fields and their (orientation-reversal) duals}
\nn{remark that if top dimensional fields are not already linear
then we will soon linearize them(?)}
-The definition of a system of fields is intended to generalize
+The definition of a system of fields is intended to generalize
the relevant properties of the following two examples of fields.
The first example: Fix a target space $B$ and define $\cC(X)$ (where $X$
-is a manifold of dimension $n$ or less) to be the set of
+is a manifold of dimension $n$ or less) to be the set of
all maps from $X$ to $B$.
The second example will take longer to explain.
-Given an $n$-category $C$ with the right sort of duality
-(e.g. pivotal 2-category, 1-category with duals, star 1-category, disklike $n$-category),
+Given an $n$-category $C$ with the right sort of duality
+(e.g. pivotal 2-category, 1-category with duals, star 1-category, disklike $n$-category),
we can construct a system of fields as follows.
Roughly speaking, $\cC(X)$ will the set of all embedded cell complexes in $X$
with codimension $i$ cells labeled by $i$-morphisms of $C$.
@@ -149,18 +143,18 @@
an object (0-morphism) of the 1-category $C$.
A field on a 1-manifold $S$ consists of
\begin{itemize}
- \item A cell decomposition of $S$ (equivalently, a finite collection
+ \item A cell decomposition of $S$ (equivalently, a finite collection
of points in the interior of $S$);
- \item a labeling of each 1-cell (and each half 1-cell adjacent to $\bd S$)
+ \item a labeling of each 1-cell (and each half 1-cell adjacent to $\bd S$)
by an object (0-morphism) of $C$;
- \item a transverse orientation of each 0-cell, thought of as a choice of
+ \item a transverse orientation of each 0-cell, thought of as a choice of
``domain" and ``range" for the two adjacent 1-cells; and
- \item a labeling of each 0-cell by a morphism (1-morphism) of $C$, with
+ \item a labeling of each 0-cell by a morphism (1-morphism) of $C$, with
domain and range determined by the transverse orientation and the labelings of the 1-cells.
\end{itemize}
If $C$ is an algebra (i.e. if $C$ has only one 0-morphism) we can ignore the labels
-of 1-cells, so a field on a 1-manifold $S$ is a finite collection of points in the
+of 1-cells, so a field on a 1-manifold $S$ is a finite collection of points in the
interior of $S$, each transversely oriented and each labeled by an element (1-morphism)
of the algebra.
@@ -175,19 +169,19 @@
A field of a 1-manifold is defined as in the $n=1$ case, using the 0- and 1-morphisms of $C$.
A field on a 2-manifold $Y$ consists of
\begin{itemize}
- \item A cell decomposition of $Y$ (equivalently, a graph embedded in $Y$ such
+ \item A cell decomposition of $Y$ (equivalently, a graph embedded in $Y$ such
that each component of the complement is homeomorphic to a disk);
- \item a labeling of each 2-cell (and each partial 2-cell adjacent to $\bd Y$)
+ \item a labeling of each 2-cell (and each partial 2-cell adjacent to $\bd Y$)
by a 0-morphism of $C$;
- \item a transverse orientation of each 1-cell, thought of as a choice of
+ \item a transverse orientation of each 1-cell, thought of as a choice of
``domain" and ``range" for the two adjacent 2-cells;
- \item a labeling of each 1-cell by a 1-morphism of $C$, with
-domain and range determined by the transverse orientation of the 1-cell
+ \item a labeling of each 1-cell by a 1-morphism of $C$, with
+domain and range determined by the transverse orientation of the 1-cell
and the labelings of the 2-cells;
- \item for each 0-cell, a homeomorphism of the boundary $R$ of a small neighborhood
+ \item for each 0-cell, a homeomorphism of the boundary $R$ of a small neighborhood
of the 0-cell to $S^1$ such that the intersections of the 1-cells with $R$ are not mapped
to $\pm 1 \in S^1$; and
- \item a labeling of each 0-cell by a 2-morphism of $C$, with domain and range
+ \item a labeling of each 0-cell by a 2-morphism of $C$, with domain and range
determined by the labelings of the 1-cells and the parameterizations of the previous
bullet.
\end{itemize}
@@ -195,10 +189,10 @@
For general $n$, a field on a $k$-manifold $X^k$ consists of
\begin{itemize}
- \item A cell decomposition of $X$;
- \item an explicit general position homeomorphism from the link of each $j$-cell
+ \item A cell decomposition of $X$;
+ \item an explicit general position homeomorphism from the link of each $j$-cell
to the boundary of the standard $(k-j)$-dimensional bihedron; and
- \item a labeling of each $j$-cell by a $(k-j)$-dimensional morphism of $C$, with
+ \item a labeling of each $j$-cell by a $(k-j)$-dimensional morphism of $C$, with
domain and range determined by the labelings of the link of $j$-cell.
\end{itemize}
@@ -208,10 +202,10 @@
\medskip
-For top dimensional ($n$-dimensional) manifolds, we're actually interested
+For top dimensional ($n$-dimensional) manifolds, we're actually interested
in the linearized space of fields.
By default, define $\cC_l(X) = \c[\cC(X)]$; that is, $\cC_l(X)$ is
-the vector space of finite
+the vector space of finite
linear combinations of fields on $X$.
If $X$ has boundary, we of course fix a boundary condition: $\cC_l(X; a) = \c[\cC(X; a)]$.
Thus the restriction (to boundary) maps are well defined because we never
@@ -220,9 +214,9 @@
In some cases we don't linearize the default way; instead we take the
spaces $\cC_l(X; a)$ to be part of the data for the system of fields.
In particular, for fields based on linear $n$-category pictures we linearize as follows.
-Define $\cC_l(X; a) = \c[\cC(X; a)]/K$, where $K$ is the space generated by
+Define $\cC_l(X; a) = \c[\cC(X; a)]/K$, where $K$ is the space generated by
obvious relations on 0-cell labels.
-More specifically, let $L$ be a cell decomposition of $X$
+More specifically, let $L$ be a cell decomposition of $X$
and let $p$ be a 0-cell of $L$.
Let $\alpha_c$ and $\alpha_d$ be two labelings of $L$ which are identical except that
$\alpha_c$ labels $p$ by $c$ and $\alpha_d$ labels $p$ by $d$.
@@ -231,9 +225,9 @@
to infer the meaning of $\alpha_{\lambda c + d}$.
Note that we are still assuming that $n$-categories have linear spaces of $n$-morphisms.
-\nn{Maybe comment further: if there's a natural basis of morphisms, then no need;
+\nn{Maybe comment further: if there's a natural basis of morphisms, then no need;
will do something similar below; in general, whenever a label lives in a linear
-space we do something like this; ? say something about tensor
+space we do something like this; ? say something about tensor
product of all the linear label spaces? Yes:}
For top dimensional ($n$-dimensional) manifolds, we linearize as follows.
@@ -243,7 +237,7 @@
space determined by the labeling of the link of the 0-cell.
(If the 0-cell were labeled, the label would live in this space.)
We associate to each almost-labeling the tensor product of these spaces (one for each 0-cell).
-We now define $\cC_l(X; a)$ to be the direct sum over all almost labelings of the
+We now define $\cC_l(X; a)$ to be the direct sum over all almost labelings of the
above tensor products.
@@ -251,12 +245,12 @@
\subsection{Local relations}
Let $B^n$ denote the standard $n$-ball.
-A {\it local relation} is a collection subspaces $U(B^n; c) \sub \cC_l(B^n; c)$
+A {\it local relation} is a collection subspaces $U(B^n; c) \sub \cC_l(B^n; c)$
(for all $c \in \cC(\bd B^n)$) satisfying the following (three?) properties.
-\nn{Roughly, these are (1) the local relations imply (extended) isotopy;
+\nn{Roughly, these are (1) the local relations imply (extended) isotopy;
(2) $U(B^n; \cdot)$ is an ideal w.r.t.\ gluing; and
-(3) this ideal is generated by ``small" generators (contained in an open cover of $B^n$).
+(3) this ideal is generated by ``small" generators (contained in an open cover of $B^n$).
See KW TQFT notes for details. Need to transfer details to here.}
For maps into spaces, $U(B^n; c)$ is generated by things of the form $a-b \in \cC_l(B^n; c)$,
@@ -292,7 +286,7 @@
In this section we will usually suppress boundary conditions on $X$ from the notation
(e.g. write $\cC_l(X)$ instead of $\cC_l(X; c)$).
-We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0
+We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0
submanifold of $X$, then $X \setmin Y$ implicitly means the closure
$\overline{X \setmin Y}$.
@@ -326,7 +320,7 @@
There is a map $\bd : \bc_1(X) \to \bc_0(X)$ which sends $(B, r, u)$ to $ru$, the linear
combination of fields on $X$ obtained by gluing $r$ to $u$.
-In other words $\bd : \bc_1(X) \to \bc_0(X)$ is given by
+In other words $\bd : \bc_1(X) \to \bc_0(X)$ is given by
just erasing the blob from the picture
(but keeping the blob label $u$).
@@ -334,7 +328,7 @@
is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$.
$\bc_2(X)$ is the space of all relations (redundancies) among the relations of $\bc_1(X)$.
-More specifically, $\bc_2(X)$ is the space of all finite linear combinations of
+More specifically, $\bc_2(X)$ is the space of all finite linear combinations of
2-blob diagrams (defined below), modulo the usual linear label relations.
\nn{and also modulo blob reordering relations?}
@@ -403,14 +397,14 @@
Let $x_a$ be the blob diagram with label $a$, and define $x_b$ and $x_c$ similarly.
Then we impose the relation
\eq{
- x_c = \lambda x_a + x_b .
+ x_c = \lambda x_a + x_b .
}
\nn{should do this in terms of direct sums of tensor products}
Let $x$ and $x'$ be two blob diagrams which differ only by a permutation $\pi$
of their blob labelings.
Then we impose the relation
\eq{
- x = \sign(\pi) x' .
+ x = \sign(\pi) x' .
}
(Alert readers will have noticed that for $k=2$ our definition
@@ -430,7 +424,7 @@
where $r_l$ is the restriction of $r$ to $B_l \setmin B_j$.
Finally, define
\eq{
- \bd(b) = \sum_{j=0}^{k-1} (-1)^j E_j(b).
+ \bd(b) = \sum_{j=0}^{k-1} (-1)^j E_j(b).
}
The $(-1)^j$ factors imply that the terms of $\bd^2(b)$ all cancel.
Thus we have a chain complex.
@@ -438,8 +432,8 @@
\nn{?? say something about the ``shape" of tree? (incl = cone, disj = product)}
-\nn{TO DO:
-expand definition to handle DGA and $A_\infty$ versions of $n$-categories;
+\nn{TO DO:
+expand definition to handle DGA and $A_\infty$ versions of $n$-categories;
relations to Chas-Sullivan string stuff}
@@ -451,7 +445,7 @@
\end{prop}
\begin{proof}
Given blob diagrams $b_1$ on $X$ and $b_2$ on $Y$, we can combine them
-(putting the $b_1$ blobs before the $b_2$ blobs in the ordering) to get a
+(putting the $b_1$ blobs before the $b_2$ blobs in the ordering) to get a
blob diagram $(b_1, b_2)$ on $X \du Y$.
Because of the blob reordering relations, all blob diagrams on $X \du Y$ arise this way.
In the other direction, any blob diagram on $X\du Y$ is equal (up to sign)
@@ -467,8 +461,8 @@
For the next proposition we will temporarily restore $n$-manifold boundary
conditions to the notation.
-Suppose that for all $c \in \cC(\bd B^n)$
-we have a splitting $s: H_0(\bc_*(B^n, c)) \to \bc_0(B^n; c)$
+Suppose that for all $c \in \cC(\bd B^n)$
+we have a splitting $s: H_0(\bc_*(B^n, c)) \to \bc_0(B^n; c)$
of the quotient map
$p: \bc_0(B^n; c) \to H_0(\bc_*(B^n, c))$.
\nn{always the case if we're working over $\c$}.
@@ -490,7 +484,7 @@
\nn{$x$ is a 0-blob diagram, i.e. $x \in \cC(B^n; c)$}
\end{proof}
-(Note that for the above proof to work, we need the linear label relations
+(Note that for the above proof to work, we need the linear label relations
for blob labels.
Also we need to blob reordering relations (?).)
@@ -525,7 +519,7 @@
\begin{prop}
For fixed fields ($n$-cat), $\bc_*$ is a functor from the category
-of $n$-manifolds and diffeomorphisms to the category of chain complexes and
+of $n$-manifolds and diffeomorphisms to the category of chain complexes and
(chain map) isomorphisms.
\qed
\end{prop}
@@ -558,9 +552,9 @@
\begin{prop}
There is a natural chain map
\eq{
- \gl: \bigoplus_a \bc_*(X; a, -a, c) \to \bc_*(X\sgl; c\sgl).
+ \gl: \bigoplus_a \bc_*(X; a, -a, c) \to \bc_*(X\sgl; c\sgl).
}
-The sum is over all fields $a$ on $Y$ compatible at their
+The sum is over all fields $a$ on $Y$ compatible at their
($n{-}2$-dimensional) boundaries with $c$.
`Natural' means natural with respect to the actions of diffeomorphisms.
\qed
@@ -574,7 +568,7 @@
(Typically one of $X_1$ or $X_2$ is a disjoint union of balls.)
For $x_i \in \bc_*(X_i)$, we introduce the notation
\eq{
- x_1 \bullet x_2 \deq \gl(x_1 \otimes x_2) .
+ x_1 \bullet x_2 \deq \gl(x_1 \otimes x_2) .
}
Note that we have resumed our habit of omitting boundary labels from the notation.
@@ -589,17 +583,17 @@
\section{$n=1$ and Hochschild homology}
In this section we analyze the blob complex in dimension $n=1$
-and find that for $S^1$ the homology of the blob complex is the
+and find that for $S^1$ the homology of the blob complex is the
Hochschild homology of the category (algebroid) that we started with.
\nn{or maybe say here that the complexes are quasi-isomorphic? in general,
should perhaps put more emphasis on the complexes and less on the homology.}
Notation: $HB_i(X) = H_i(\bc_*(X))$.
-Let us first note that there is no loss of generality in assuming that our system of
+Let us first note that there is no loss of generality in assuming that our system of
fields comes from a category.
(Or maybe (???) there {\it is} a loss of generality.
-Given any system of fields, $A(I; a, b) = \cC(I; a, b)/U(I; a, b)$ can be
+Given any system of fields, $A(I; a, b) = \cC(I; a, b)/U(I; a, b)$ can be
thought of as the morphisms of a 1-category $C$.
More specifically, the objects of $C$ are $\cC(pt)$, the morphisms from $a$ to $b$
are $A(I; a, b)$, and composition is given by gluing.
@@ -624,7 +618,7 @@
\begin{itemize}
\item $\cC(pt) = \ob(C)$ .
\item Let $R$ be a 1-manifold and $c \in \cC(\bd R)$.
-Then an element of $\cC(R; c)$ is a collection of (transversely oriented)
+Then an element of $\cC(R; c)$ is a collection of (transversely oriented)
points in the interior
of $R$, each labeled by a morphism of $C$.
The intervals between the points are labeled by objects of $C$, consistent with
@@ -635,12 +629,12 @@
the same way.
\item For $x \in \mor(a, b)$ let $\chi(x) \in \cC(I; a, b)$ be the field with a single
point (at some standard location) labeled by $x$.
-Then the kernel of the evaluation map $U(I; a, b)$ is generated by things of the
+Then the kernel of the evaluation map $U(I; a, b)$ is generated by things of the
form $y - \chi(e(y))$.
Thus we can, if we choose, restrict the blob twig labels to things of this form.
\end{itemize}
-We want to show that $HB_*(S^1)$ is naturally isomorphic to the
+We want to show that $HB_*(S^1)$ is naturally isomorphic to the
Hochschild homology of $C$.
\nn{Or better that the complexes are homotopic
or quasi-isomorphic.}
@@ -691,13 +685,13 @@
First we show that $F_*(C\otimes C)$ is
quasi-isomorphic to the 0-step complex $C$.
-Let $F'_* \sub F_*(C\otimes C)$ be the subcomplex where the label of
+Let $F'_* \sub F_*(C\otimes C)$ be the subcomplex where the label of
the point $*$ is $1 \otimes 1 \in C\otimes C$.
We will show that the inclusion $i: F'_* \to F_*(C\otimes C)$ is a quasi-isomorphism.
Fix a small $\ep > 0$.
Let $B_\ep$ be the ball of radius $\ep$ around $* \in S^1$.
-Let $F^\ep_* \sub F_*(C\otimes C)$ be the subcomplex
+Let $F^\ep_* \sub F_*(C\otimes C)$ be the subcomplex
generated by blob diagrams $b$ such that $B_\ep$ is either disjoint from
or contained in each blob of $b$, and the two boundary points of $B_\ep$ are not labeled points of $b$.
For a field (picture) $y$ on $B_\ep$, let $s_\ep(y)$ be the equivalent picture with~$*$
@@ -712,7 +706,7 @@
$x$ as a new twig blob, with label $y - s_\ep(y)$, where $y$ is the restriction of $x$ to $B_\ep$.
If $*$ is contained in a twig blob $B$ with label $u = \sum z_i$, $j_\ep(x)$ is obtained as follows.
Let $y_i$ be the restriction of $z_i$ to $B_\ep$.
-Let $x_i$ be equal to $x$ outside of $B$, equal to $z_i$ on $B \setmin B_\ep$,
+Let $x_i$ be equal to $x$ outside of $B$, equal to $z_i$ on $B \setmin B_\ep$,
and have an additional blob $B_\ep$ with label $y_i - s_\ep(y_i)$.
Define $j_\ep(x) = \sum x_i$.
\nn{need to check signs coming from blob complex differential}
@@ -721,7 +715,7 @@
The key property of $j_\ep$ is
\eq{
- \bd j_\ep + j_\ep \bd = \id - \sigma_\ep ,
+ \bd j_\ep + j_\ep \bd = \id - \sigma_\ep ,
}
where $\sigma_\ep : F^\ep_* \to F^\ep_*$ is given by replacing the restriction $y$ of each field
mentioned in $x \in F^\ep_*$ with $s_\ep(y)$.
@@ -731,10 +725,10 @@
is a homotopy inverse to the inclusion $F'_* \to F_*(C\otimes C)$.
One strategy would be to try to stitch together various $j_\ep$ for progressively smaller
$\ep$ and show that $F'_*$ is homotopy equivalent to $F_*(C\otimes C)$.
-Instead, we'll be less ambitious and just show that
+Instead, we'll be less ambitious and just show that
$F'_*$ is quasi-isomorphic to $F_*(C\otimes C)$.
-If $x$ is a cycle in $F_*(C\otimes C)$, then for sufficiently small $\ep$ we have
+If $x$ is a cycle in $F_*(C\otimes C)$, then for sufficiently small $\ep$ we have
$x \in F_*^\ep$.
(This is true for any chain in $F_*(C\otimes C)$, since chains are sums of
finitely many blob diagrams.)
@@ -743,7 +737,7 @@
If $y \in F_*(C\otimes C)$ and $\bd y = x \in F'_*$, then $y \in F^\ep_*$ for some $\ep$
and
\eq{
- \bd y = \bd (\sigma_\ep(y) + j_\ep(x)) .
+ \bd y = \bd (\sigma_\ep(y) + j_\ep(x)) .
}
Since $\sigma_\ep(y) + j_\ep(x) \in F'$, it follows that the inclusion map is injective on homology.
This completes the proof that $F'_*$ is quasi-isomorphic to $F_*(C\otimes C)$.
@@ -769,7 +763,7 @@
Next we construct a degree 1 map (homotopy) $h: F'_* \to F'_*$ such that
for all $x \in F'_*$ we have
\eq{
- x - \bd h(x) - h(\bd x) \in F''_* .
+ x - \bd h(x) - h(\bd x) \in F''_* .
}
Since $F'_0 = F''_0$, we can take $h_0 = 0$.
Let $x \in F'_1$, with single blob $B \sub S^1$.
@@ -793,7 +787,7 @@
Finally, we show that $F''_*$ is contractible.
\nn{need to also show that $H_0$ is the right thing; easy, but I won't do it now}
Let $x$ be a cycle in $F''_*$.
-The union of the supports of the diagrams in $x$ does not contain $*$, so there exists a
+The union of the supports of the diagrams in $x$ does not contain $*$, so there exists a
ball $B \subset S^1$ containing the union of the supports and not containing $*$.
Adding $B$ as a blob to $x$ gives a contraction.
\nn{need to say something else in degree zero}
@@ -813,11 +807,11 @@
* is a labeled point in $y$.
Otherwise, define $s(y)$ to be the result of adding a label 1 (identity morphism) at *.
Let $x \in \bc_*(S^1)$.
-Let $s(x)$ be the result of replacing each field $y$ (containing *) mentioned in
+Let $s(x)$ be the result of replacing each field $y$ (containing *) mentioned in
$x$ with $y$.
It is easy to check that $s$ is a chain map and $s \circ i = \id$.
-Let $G^\ep_* \sub \bc_*(S^1)$ be the subcomplex where there are no labeled points
+Let $G^\ep_* \sub \bc_*(S^1)$ be the subcomplex where there are no labeled points
in a neighborhood $B_\ep$ of *, except perhaps *.
Note that for any chain $x \in \bc_*(S^1)$, $x \in G^\ep_*$ for sufficiently small $\ep$.
\nn{rest of argument goes similarly to above}
@@ -833,8 +827,36 @@
Probably it's worth writing down an explicit map even if we don't need to.}
+We can also describe explicitly a map from the standard Hochschild
+complex to the blob complex on the circle. \nn{What properties does this
+map have?}
+\begin{figure}%
+$$\mathfig{0.6}{barycentric/barycentric}$$
+\caption{The Hochschild chain $a \tensor b \tensor c$ is sent to
+the sum of six blob $2$-chains, corresponding to a barycentric subdivision of a $2$-simplex.}
+\label{fig:Hochschild-example}%
+\end{figure}
+As an example, Figure \ref{fig:Hochschild-example} shows the image of the Hochschild chain $a \tensor b \tensor c$. Only the $0$-cells are shown explicitly.
+The edges marked $x, y$ and $z$ carry the $1$-chains
+\begin{align*}
+x & = \mathfig{0.1}{barycentric/ux} & u_x = \mathfig{0.1}{barycentric/ux_ca} - \mathfig{0.1}{barycentric/ux_c-a} \\
+y & = \mathfig{0.1}{barycentric/uy} & u_y = \mathfig{0.1}{barycentric/uy_cab} - \mathfig{0.1}{barycentric/uy_ca-b} \\
+z & = \mathfig{0.1}{barycentric/uz} & u_z = \mathfig{0.1}{barycentric/uz_c-a-b} - \mathfig{0.1}{barycentric/uz_cab}
+\end{align*}
+and the $2$-chain labelled $A$ is
+\begin{equation*}
+A = \mathfig{0.1}{barycentric/Ax}+\mathfig{0.1}{barycentric/Ay}.
+\end{equation*}
+Note that we then have
+\begin{equation*}
+\bdy A = x+y+z.
+\end{equation*}
+
+In general, the Hochschild chain $\Tensor_{i=1}^n a_i$ is sent to the sum of $n!$ blob $(n-1)$-chains, indexed by permutations,
+$$\phi\left(\Tensor_{i=1}^n a_i) = \sum_{\pi} \phi^\pi(a_1, \ldots, a_n)$$
+with ...
\section{Action of $C_*(\Diff(X))$} \label{diffsect}
@@ -849,16 +871,16 @@
\begin{prop} \label{CDprop}
For each $n$-manifold $X$ there is a chain map
\eq{
- e_X : CD_*(X) \otimes \bc_*(X) \to \bc_*(X) .
+ e_X : CD_*(X) \otimes \bc_*(X) \to \bc_*(X) .
}
On $CD_0(X) \otimes \bc_*(X)$ it agrees with the obvious action of $\Diff(X)$ on $\bc_*(X)$
(Proposition (\ref{diff0prop})).
For any splitting $X = X_1 \cup X_2$, the following diagram commutes
\eq{ \xymatrix{
- CD_*(X) \otimes \bc_*(X) \ar[r]^{e_X} & \bc_*(X) \\
- CD_*(X_1) \otimes CD_*(X_2) \otimes \bc_*(X_1) \otimes \bc_*(X_2)
- \ar@/_4ex/[r]_{e_{X_1} \otimes e_{X_2}} \ar[u]^{\gl \otimes \gl} &
- \bc_*(X_1) \otimes \bc_*(X_2) \ar[u]_{\gl}
+ CD_*(X) \otimes \bc_*(X) \ar[r]^{e_X} & \bc_*(X) \\
+ CD_*(X_1) \otimes CD_*(X_2) \otimes \bc_*(X_1) \otimes \bc_*(X_2)
+ \ar@/_4ex/[r]_{e_{X_1} \otimes e_{X_2}} \ar[u]^{\gl \otimes \gl} &
+ \bc_*(X_1) \otimes \bc_*(X_2) \ar[u]_{\gl}
} }
Any other map satisfying the above two properties is homotopic to $e_X$.
\end{prop}
@@ -876,18 +898,18 @@
A $k$-parameter family of diffeomorphisms $f: P \times X \to X$ is
{\it adapted to $\cU$} if there is a factorization
\eq{
- P = P_1 \times \cdots \times P_m
+ P = P_1 \times \cdots \times P_m
}
(for some $m \le k$)
and families of diffeomorphisms
\eq{
- f_i : P_i \times X \to X
+ f_i : P_i \times X \to X
}
-such that
+such that
\begin{itemize}
\item each $f_i(p, \cdot): X \to X$ is supported on some connected $V_i \sub X$;
\item the $V_i$'s are mutually disjoint;
-\item each $V_i$ is the union of at most $k_i$ of the $U_\alpha$'s,
+\item each $V_i$ is the union of at most $k_i$ of the $U_\alpha$'s,
where $k_i = \dim(P_i)$; and
\item $f(p, \cdot) = f_1(p_1, \cdot) \circ \cdots \circ f_m(p_m, \cdot) \circ g$
for all $p = (p_1, \ldots, p_m)$, for some fixed $g \in \Diff(X)$.
@@ -904,12 +926,12 @@
\medskip
-Let $B_1, \ldots, B_m$ be a collection of disjoint balls in $X$
+Let $B_1, \ldots, B_m$ be a collection of disjoint balls in $X$
(e.g.~the support of a blob diagram).
We say that $f:P\times X\to X$ is {\it compatible} with $\{B_j\}$ if
$f$ has support a disjoint collection of balls $D_i \sub X$ and for all $i$ and $j$
either $B_j \sub D_i$ or $B_j \cap D_i = \emptyset$.
-A chain $x \in CD_k(X)$ is compatible with $\{B_j\}$ if it is a sum of singular cells,
+A chain $x \in CD_k(X)$ is compatible with $\{B_j\}$ if it is a sum of singular cells,
each of which is compatible.
(Note that we could strengthen the definition of compatibility to incorporate
a factorization condition, similar to the definition of ``adapted to" above.
@@ -920,14 +942,14 @@
Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is compatible with $\{B_j\}$.
\end{cor}
\begin{proof}
-This will follow from Lemma \ref{extension_lemma} for
+This will follow from Lemma \ref{extension_lemma} for
appropriate choice of cover $\cU = \{U_\alpha\}$.
Let $U_{\alpha_1}, \ldots, U_{\alpha_k}$ be any $k$ open sets of $\cU$, and let
$V_1, \ldots, V_m$ be the connected components of $U_{\alpha_1}\cup\cdots\cup U_{\alpha_k}$.
Choose $\cU$ fine enough so that there exist disjoint balls $B'_j \sup B_j$ such that for all $i$ and $j$
either $V_i \sub B'_j$ or $V_i \cap B'_j = \emptyset$.
-Apply Lemma \ref{extension_lemma} first to each singular cell $f_i$ of $\bd x$,
+Apply Lemma \ref{extension_lemma} first to each singular cell $f_i$ of $\bd x$,
with the (compatible) support of $f_i$ in place of $X$.
This insures that the resulting homotopy $h_i$ is compatible.
Now apply Lemma \ref{extension_lemma} to $x + \sum h_i$.
@@ -957,7 +979,7 @@
\nn{not sure what the best way to deal with boundary is; for now just give main argument, worry
about boundary later}
-Recall that we are given
+Recall that we are given
an open cover $\cU = \{U_\alpha\}$ and an
$x \in CD_k(X)$ such that $\bd x$ is adapted to $\cU$.
We must find a homotopy of $x$ (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$.
@@ -965,20 +987,20 @@
Let $\{r_\alpha : X \to [0,1]\}$ be a partition of unity for $\cU$.
As a first approximation to the argument we will eventually make, let's replace $x$
-with a single singular cell
+with a single singular cell
\eq{
- f: P \times X \to X .
+ f: P \times X \to X .
}
Also, we'll ignore for now issues around $\bd P$.
Our homotopy will have the form
\eqar{
- F: I \times P \times X &\to& X \\
- (t, p, x) &\mapsto& f(u(t, p, x), x)
+ F: I \times P \times X &\to& X \\
+ (t, p, x) &\mapsto& f(u(t, p, x), x)
}
for some function
\eq{
- u : I \times P \times X \to P .
+ u : I \times P \times X \to P .
}
First we describe $u$, then we argue that it does what we want it to do.
@@ -1007,16 +1029,16 @@
For $p \in D$ we define
\eq{
- u(t, p, x) = (1-t)p + t \sum_\alpha r_\alpha(x) p_{c_\alpha} .
+ u(t, p, x) = (1-t)p + t \sum_\alpha r_\alpha(x) p_{c_\alpha} .
}
(Recall that $P$ is a single linear cell, so the weighted average of points of $P$
makes sense.)
So far we have defined $u(t, p, x)$ when $p$ lies in a $k$-handle of $J$.
-For handles of $J$ of index less than $k$, we will define $u$ to
+For handles of $J$ of index less than $k$, we will define $u$ to
interpolate between the values on $k$-handles defined above.
-If $p$ lies in a $k{-}1$-handle $E$, let $\eta : E \to [0,1]$ be the normal coordinate
+If $p$ lies in a $k{-}1$-handle $E$, let $\eta : E \to [0,1]$ be the normal coordinate
of $E$.
In particular, $\eta$ is equal to 0 or 1 only at the intersection of $E$
with a $k$-handle.
@@ -1026,8 +1048,8 @@
adjacent to the $k{-}1$-cell corresponding to $E$.
For $p \in E$, define
\eq{
- u(t, p, x) = (1-t)p + t \left( \sum_{\alpha \ne \beta} r_\alpha(x) p_{c_\alpha}
- + r_\beta(x) (\eta(p) q_1 + (1-\eta(p)) q_0) \right) .
+ u(t, p, x) = (1-t)p + t \left( \sum_{\alpha \ne \beta} r_\alpha(x) p_{c_\alpha}
+ + r_\beta(x) (\eta(p) q_1 + (1-\eta(p)) q_0) \right) .
}
In general, for $E$ a $k{-}j$-handle, there is a normal coordinate
@@ -1040,10 +1062,10 @@
For each $\beta \in \cN$, let $\{q_{\beta i}\}$ be the set of points in $P$ associated to the aforementioned $k$-cells.
Now define, for $p \in E$,
\eq{
- u(t, p, x) = (1-t)p + t \left(
- \sum_{\alpha \notin \cN} r_\alpha(x) p_{c_\alpha}
- + \sum_{\beta \in \cN} r_\beta(x) \left( \sum_i \eta_{\beta i}(p) \cdot q_{\beta i} \right)
- \right) .
+ u(t, p, x) = (1-t)p + t \left(
+ \sum_{\alpha \notin \cN} r_\alpha(x) p_{c_\alpha}
+ + \sum_{\beta \in \cN} r_\beta(x) \left( \sum_i \eta_{\beta i}(p) \cdot q_{\beta i} \right)
+ \right) .
}
Here $\eta_{\beta i}(p)$ is the weight given to $q_{\beta i}$ by the linear extension
mentioned above.
@@ -1062,8 +1084,8 @@
We must show that the derivative $\pd{F}{x}(t, p, x)$ is non-singular for all $(t, p, x)$.
We have
\eq{
-% \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) .
- \pd{F}{x} = \pd{f}{x} + \pd{f}{p} \pd{u}{x} .
+% \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) .
+ \pd{F}{x} = \pd{f}{x} + \pd{f}{p} \pd{u}{x} .
}
Since $f$ is a family of diffeomorphisms, $\pd{f}{x}$ is non-singular and
\nn{bounded away from zero, or something like that}.
@@ -1083,7 +1105,7 @@
This will complete the proof of the lemma.
\nn{except for boundary issues and the `$P$ is a cell' assumption}
-Let $j$ be the codimension of $D$.
+Let $j$ be the codimension of $D$.
(Or rather, the codimension of its corresponding cell. From now on we will not make a distinction
between handle and corresponding cell.)
Then $j = j_1 + \cdots + j_m$, $0 \le m \le k$,
@@ -1110,7 +1132,7 @@
The singular 1-cell is supported on $U_\beta$, since $r_\beta = 0$ outside of this set.
Next case: $j=2$, $m=1$, $j_1 = 2$.
-This is similar to the previous case, except that the normal bundle is 2-dimensional instead of
+This is similar to the previous case, except that the normal bundle is 2-dimensional instead of
1-dimensional.
We have that $D \to \Diff(X)$ is a product of a constant singular $k{-}2$-cell
and a 2-cell with support $U_\beta$.
@@ -1136,15 +1158,15 @@
\section{$A_\infty$ action on the boundary}
Let $Y$ be an $n{-}1$-manifold.
-The collection of complexes $\{\bc_*(Y\times I; a, b)\}$, where $a, b \in \cC(Y)$ are boundary
+The collection of complexes $\{\bc_*(Y\times I; a, b)\}$, where $a, b \in \cC(Y)$ are boundary
conditions on $\bd(Y\times I) = Y\times \{0\} \cup Y\times\{1\}$, has the structure
of an $A_\infty$ category.
Composition of morphisms (multiplication) depends of a choice of homeomorphism
$I\cup I \cong I$. Given this choice, gluing gives a map
\eq{
- \bc_*(Y\times I; a, b) \otimes \bc_*(Y\times I; b, c) \to \bc_*(Y\times (I\cup I); a, c)
- \cong \bc_*(Y\times I; a, c)
+ \bc_*(Y\times I; a, b) \otimes \bc_*(Y\times I; b, c) \to \bc_*(Y\times (I\cup I); a, c)
+ \cong \bc_*(Y\times I; a, c)
}
Using (\ref{CDprop}) and the inclusion $\Diff(I) \sub \Diff(Y\times I)$ gives the various
higher associators of the $A_\infty$ structure, more or less canonically.
@@ -1155,7 +1177,7 @@
Similarly, if $Y \sub \bd X$, a choice of collaring homeomorphism
$(Y\times I) \cup_Y X \cong X$ gives the collection of complexes $\bc_*(X; r, a)$
-(variable $a \in \cC(Y)$; fixed $r \in \cC(\bd X \setmin Y)$) the structure of a representation of the
+(variable $a \in \cC(Y)$; fixed $r \in \cC(\bd X \setmin Y)$) the structure of a representation of the
$A_\infty$ category $\{\bc_*(Y\times I; \cdot, \cdot)\}$.
Again the higher associators come from the action of $\Diff(I)$ on a collar neighborhood
of $Y$ in $X$.
@@ -1176,14 +1198,14 @@
Gluing the two copies of $Y$ together we obtain a new $n$-manifold $X\sgl$.
We wish to describe the blob complex of $X\sgl$ in terms of the blob complex
of $X$.
-More precisely, we want to describe $\bc_*(X\sgl; c\sgl)$,
+More precisely, we want to describe $\bc_*(X\sgl; c\sgl)$,
where $c\sgl \in \cC(\bd X\sgl)$,
in terms of the collection $\{\bc_*(X; c, \cdot, \cdot)\}$, thought of as a representation
of the $A_\infty$ category $\cA(Y\du-Y) \cong \cA(Y)\times \cA(Y)\op$.
\begin{thm}
$\bc_*(X\sgl; c\sgl)$ is quasi-isomorphic to the the self tensor product
-of $\{\bc_*(X; c, \cdot, \cdot)\}$ over $\cA(Y)$.
+of $\{\bc_*(X; c, \cdot, \cdot)\}$ over $\cA(Y)$.
\end{thm}
The proof will occupy the remainder of this section.
@@ -1200,7 +1222,7 @@
\section{Extension to ...}
-\nn{Need to let the input $n$-category $C$ be a graded thing
+\nn{Need to let the input $n$-category $C$ be a graded thing
(e.g.~DGA or $A_\infty$ $n$-category).}
\nn{maybe this should be done earlier in the exposition?
@@ -1233,6 +1255,3 @@
%$m: \bc_0(B^n; c, c') \to \mor(c, c')$.
%If we regard $\mor(c, c')$ as a complex concentrated in degree 0, then this becomes a chain
%map $m: \bc_*(B^n; c, c') \to \mor(c, c')$.
-
-
-