diff -r a3311a926113 -r c5a43be00ed4 blob1.tex --- a/blob1.tex Tue Jul 21 18:45:18 2009 +0000 +++ b/blob1.tex Wed Jul 22 03:38:13 2009 +0000 @@ -1,24 +1,20 @@ \documentclass[11pt,leqno]{amsart} \newcommand{\pathtotrunk}{./} -\input{text/article_preamble.tex} -\input{text/top_matter.tex} -\input{text/kw_macros.tex} - +\input{text/article_preamble} +\input{text/top_matter} +\input{text/kw_macros} \title{Blob Homology} \begin{document} - - \makeatletter \@addtoreset{equation}{section} \gdef\theequation{\thesection.\arabic{equation}} \makeatother - \maketitle \textbf{Draft version, do not distribute.} @@ -65,888 +61,32 @@ -\input{text/intro.tex} - - -\section{Definitions} -\label{sec:definitions} - -\subsection{Systems of fields} -\label{sec:fields} - -Let $\cM_k$ denote the category (groupoid, in fact) with objects -oriented PL manifolds of dimension -$k$ and morphisms homeomorphisms. -(We could equally well work with a different category of manifolds --- -unoriented, topological, smooth, spin, etc. --- but for definiteness we -will stick with oriented PL.) - -Fix a top dimension $n$, and a symmetric monoidal category $\cS$ whose objects are sets. While reading the definition, you should just think about the case $\cS = \Set$ with cartesian product, until you reach the discussion of a \emph{linear system of fields} later in this section, where $\cS = \Vect$, and \S \ref{sec:homological-fields}, where $\cS = \Kom$. - -A $n$-dimensional {\it system of fields} in $\cS$ -is a collection of functors $\cC_k : \cM_k \to \Set$ for $0 \leq k \leq n$ -together with some additional data and satisfying some additional conditions, all specified below. - -\nn{refer somewhere to my TQFT notes \cite{kw:tqft}, and possibly also to paper with Chris} - -Before finishing the definition of fields, we give two motivating examples -(actually, families of examples) of systems of fields. - -The first examples: Fix a target space $B$, and let $\cC(X)$ be the set of continuous maps -from X to $B$. - -The second examples: Fix an $n$-category $C$, and let $\cC(X)$ be -the set of sub-cell-complexes of $X$ with codimension-$j$ cells labeled by -$j$-morphisms of $C$. -One can think of such sub-cell-complexes as dual to pasting diagrams for $C$. -This is described in more detail below. - -Now for the rest of the definition of system of fields. -\begin{enumerate} -\item There are boundary restriction maps $\cC_k(X) \to \cC_{k-1}(\bd X)$, -and these maps are a natural -transformation between the functors $\cC_k$ and $\cC_{k-1}\circ\bd$. -For $c \in \cC_{k-1}(\bd X)$, we will denote by $\cC_k(X; c)$ the subset of -$\cC(X)$ which restricts to $c$. -In this context, we will call $c$ a boundary condition. -\item The subset $\cC_n(X;c)$ of top fields with a given boundary condition is an object in our symmetric monoidal category $\cS$. (This condition is of course trivial when $\cS = \Set$.) If the objects are sets with extra structure (e.g. $\cS = \Vect$ or $\Kom$), then this extra structure is considered part of the definition of $\cC_n$. Any maps mentioned below between top level fields must be morphisms in $\cS$. -\item There are orientation reversal maps $\cC_k(X) \to \cC_k(-X)$, and these maps -again comprise a natural transformation of functors. -In addition, the orientation reversal maps are compatible with the boundary restriction maps. -\item $\cC_k$ is compatible with the symmetric monoidal -structures on $\cM_k$, $\Set$ and $\cS$: $\cC_k(X \du W) \cong \cC_k(X)\times \cC_k(W)$, -compatibly with homeomorphisms, restriction to boundary, and orientation reversal. -We will call the projections $\cC(X_1 \du X_2) \to \cC(X_i)$ -restriction maps. -\item Gluing without corners. -Let $\bd X = Y \du -Y \du W$, where $Y$ and $W$ are closed $k{-}1$-manifolds. -Let $X\sgl$ denote $X$ glued to itself along $\pm Y$. -Using the boundary restriction, disjoint union, and (in one case) orientation reversal -maps, we get two maps $\cC_k(X) \to \cC(Y)$, corresponding to the two -copies of $Y$ in $\bd X$. -Let $\Eq_Y(\cC_k(X))$ denote the equalizer of these two maps. -Then (here's the axiom/definition part) there is an injective ``gluing" map -\[ - \Eq_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl) , -\] -and this gluing map is compatible with all of the above structure (actions -of homeomorphisms, boundary restrictions, orientation reversal, disjoint union). -Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity, -the gluing map is surjective. -From the point of view of $X\sgl$ and the image $Y \subset X\sgl$ of the -gluing surface, we say that fields in the image of the gluing map -are transverse to $Y$ or cuttable along $Y$. -\item Gluing with corners. -Let $\bd X = Y \cup -Y \cup W$, where $\pm Y$ and $W$ might intersect along their boundaries. -Let $X\sgl$ denote $X$ glued to itself along $\pm Y$. -Note that $\bd X\sgl = W\sgl$, where $W\sgl$ denotes $W$ glued to itself -(without corners) along two copies of $\bd Y$. -Let $c\sgl \in \cC_{k-1}(W\sgl)$ be a be a cuttable field on $W\sgl$ and let -$c \in \cC_{k-1}(W)$ be the cut open version of $c\sgl$. -Let $\cC^c_k(X)$ denote the subset of $\cC(X)$ which restricts to $c$ on $W$. -(This restriction map uses the gluing without corners map above.) -Using the boundary restriction, gluing without corners, and (in one case) orientation reversal -maps, we get two maps $\cC^c_k(X) \to \cC(Y)$, corresponding to the two -copies of $Y$ in $\bd X$. -Let $\Eq^c_Y(\cC_k(X))$ denote the equalizer of these two maps. -Then (here's the axiom/definition part) there is an injective ``gluing" map -\[ - \Eq^c_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl, c\sgl) , -\] -and this gluing map is compatible with all of the above structure (actions -of homeomorphisms, boundary restrictions, orientation reversal, disjoint union). -Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity, -the gluing map is surjective. -From the point of view of $X\sgl$ and the image $Y \subset X\sgl$ of the -gluing surface, we say that fields in the image of the gluing map -are transverse to $Y$ or cuttable along $Y$. -\item There are maps $\cC_{k-1}(Y) \to \cC_k(Y \times I)$, denoted -$c \mapsto c\times I$. -These maps comprise a natural transformation of functors, and commute appropriately -with all the structure maps above (disjoint union, boundary restriction, etc.). -Furthermore, if $f: Y\times I \to Y\times I$ is a fiber-preserving homeomorphism -covering $\bar{f}:Y\to Y$, then $f(c\times I) = \bar{f}(c)\times I$. -\end{enumerate} - -\nn{need to introduce two notations for glued fields --- $x\bullet y$ and $x\sgl$} - -\bigskip -Using the functoriality and $\bullet\times I$ properties above, together -with boundary collar homeomorphisms of manifolds, we can define the notion of -{\it extended isotopy}. -Let $M$ be an $n$-manifold and $Y \subset \bd M$ be a codimension zero submanifold -of $\bd M$. -Let $x \in \cC(M)$ be a field on $M$ and such that $\bd x$ is cuttable along $\bd Y$. -Let $c$ be $x$ restricted to $Y$. -Let $M \cup (Y\times I)$ denote $M$ glued to $Y\times I$ along $Y$. -Then we have the glued field $x \bullet (c\times I)$ on $M \cup (Y\times I)$. -Let $f: M \cup (Y\times I) \to M$ be a collaring homeomorphism. -Then we say that $x$ is {\it extended isotopic} to $f(x \bullet (c\times I))$. -More generally, we define extended isotopy to be the equivalence relation on fields -on $M$ generated by isotopy plus all instance of the above construction -(for all appropriate $Y$ and $x$). - -\nn{should also say something about pseudo-isotopy} - -%\bigskip -%\hrule -%\bigskip -% -%\input{text/fields.tex} -% -% -%\bigskip -%\hrule -%\bigskip - -\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(?)} - -We now describe in more detail systems of fields coming from sub-cell-complexes labeled -by $n$-category morphisms. - -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 $C$. -We'll spell this out for $n=1,2$ and then describe the general case. - -If $X$ has boundary, we require that the cell decompositions are in general -position with respect to the boundary --- the boundary intersects each cell -transversely, so cells meeting the boundary are mere half-cells. - -Put another way, the cell decompositions we consider are dual to standard cell -decompositions of $X$. - -We will always assume that our $n$-categories have linear $n$-morphisms. - -For $n=1$, a field on a 0-manifold $P$ is a labeling of each point of $P$ with -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 -of points in the interior of $S$); - \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 -``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 -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 -interior of $S$, each transversely oriented and each labeled by an element (1-morphism) -of the algebra. - -\medskip - -For $n=2$, fields are just the sort of pictures based on 2-categories (e.g.\ tensor categories) -that are common in the literature. -We describe these carefully here. - -A field on a 0-manifold $P$ is a labeling of each point of $P$ with -an object of the 2-category $C$. -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 -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 $\bd Y$) -by a 0-morphism of $C$; - \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 -and the labelings of the 2-cells; - \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 -determined by the labelings of the 1-cells and the parameterizations of the previous -bullet. -\end{itemize} -\nn{need to say this better; don't try to fit everything into the bulleted list} - -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 -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 -domain and range determined by the labelings of the link of $j$-cell. -\end{itemize} - -%\nn{next definition might need some work; I think linearity relations should -%be treated differently (segregated) from other local relations, but I'm not sure -%the next definition is the best way to do it} - -\medskip - -For top dimensional ($n$-dimensional) manifolds, we're actually interested -in the linearized space of fields. -By default, define $\lf(X) = \c[\cC(X)]$; that is, $\lf(X)$ is -the vector space of finite -linear combinations of fields on $X$. -If $X$ has boundary, we of course fix a boundary condition: $\lf(X; a) = \c[\cC(X; a)]$. -Thus the restriction (to boundary) maps are well defined because we never -take linear combinations of fields with differing boundary conditions. - -In some cases we don't linearize the default way; instead we take the -spaces $\lf(X; a)$ 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 $\lf(X; a) = \c[\cC(X; a)]/K$, where $K$ is the space generated by -obvious relations on 0-cell labels. -More specifically, let $L$ be a cell decomposition of $X$ -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 $d$. -Then the subspace $K$ is generated by things of the form -$\lambda \alpha_c + \alpha_d - \alpha_{\lambda c + d}$, where we leave it to the reader -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; -will do something similar below; in general, whenever a label lives in a linear -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. -Define an ``almost-field" to be a field without labels on the 0-cells. -(Recall that 0-cells are labeled by $n$-morphisms.) -To each unlabeled 0-cell in an almost field there corresponds a (linear) $n$-morphism -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 $\lf(X; a)$ to be the direct sum over all almost labelings of the -above tensor products. - - - -\subsection{Local relations} -\label{sec:local-relations} - - -A {\it local relation} is a collection subspaces $U(B; c) \sub \lf(B; c)$, -for all $n$-manifolds $B$ which are -homeomorphic to the standard $n$-ball and all $c \in \cC(\bd B)$, -satisfying the following properties. -\begin{enumerate} -\item functoriality: -$f(U(B; c)) = U(B', f(c))$ for all homeomorphisms $f: B \to B'$ -\item local relations imply extended isotopy: -if $x, y \in \cC(B; c)$ and $x$ is extended isotopic -to $y$, then $x-y \in U(B; c)$. -\item ideal with respect to gluing: -if $B = B' \cup B''$, $x\in U(B')$, and $c\in \cC(B'')$, then $x\bullet r \in U(B)$ -\end{enumerate} -See \cite{kw:tqft} for details. - - -For maps into spaces, $U(B; c)$ is generated by things of the form $a-b \in \lf(B; c)$, -where $a$ and $b$ are maps (fields) which are homotopic rel boundary. - -For $n$-category pictures, $U(B; c)$ is equal to the kernel of the evaluation map -$\lf(B; c) \to \mor(c', c'')$, where $(c', c'')$ is some (any) division of $c$ into -domain and range. - -\nn{maybe examples of local relations before general def?} - -Given a system of fields and local relations, we define the skein space -$A(Y^n; c)$ to be the space of all finite linear combinations of fields on -the $n$-manifold $Y$ modulo local relations. -The Hilbert space $Z(Y; c)$ for the TQFT based on the fields and local relations -is defined to be the dual of $A(Y; c)$. -(See \cite{kw:tqft} or xxxx for details.) +\input{text/intro} -\nn{should expand above paragraph} - -The blob complex is in some sense the derived version of $A(Y; c)$. - - - -\subsection{The blob complex} -\label{sec:blob-definition} - -Let $X$ be an $n$-manifold. -Assume a fixed system of fields and local relations. -In this section we will usually suppress boundary conditions on $X$ from the notation -(e.g. write $\lf(X)$ instead of $\lf(X; c)$). - -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 -$\overline{X \setmin Y}$. - -We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case $\bc_k(X)$. - -Define $\bc_0(X) = \lf(X)$. -(If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \lf(X; c)$. -We'll omit this sort of detail in the rest of this section.) -In other words, $\bc_0(X)$ is just the space of all linearized fields on $X$. - -$\bc_1(X)$ is, roughly, the space of all local relations that can be imposed on $\bc_0(X)$. -Less roughly (but still not the official definition), $\bc_1(X)$ is finite linear -combinations of 1-blob diagrams, where a 1-blob diagram to consists of -\begin{itemize} -\item An embedded closed ball (``blob") $B \sub X$. -\item A field $r \in \cC(X \setmin B; c)$ -(for some $c \in \cC(\bd B) = \cC(\bd(X \setmin B))$). -\item A local relation field $u \in U(B; c)$ -(same $c$ as previous bullet). -\end{itemize} -In order to get the linear structure correct, we (officially) define -\[ - \bc_1(X) \deq \bigoplus_B \bigoplus_c U(B; c) \otimes \lf(X \setmin B; c) . -\] -The first direct sum is indexed by all blobs $B\subset X$, and the second -by all boundary conditions $c \in \cC(\bd B)$. -Note that $\bc_1(X)$ is spanned by 1-blob diagrams $(B, u, r)$. - -Define the boundary map $\bd : \bc_1(X) \to \bc_0(X)$ by -\[ - (B, u, r) \mapsto u\bullet r, -\] -where $u\bullet r$ denotes the linear -combination of fields on $X$ obtained by gluing $u$ to $r$. -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 $u$). - -Note that the skein space $A(X)$ -is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$. - -$\bc_2(X)$ is, roughly, the space of all relations (redundancies) among the -local relations encoded in $\bc_1(X)$. -More specifically, $\bc_2(X)$ is the space of all finite linear combinations of -2-blob diagrams, of which there are two types, disjoint and nested. - -A disjoint 2-blob diagram consists of -\begin{itemize} -\item A pair of closed balls (blobs) $B_0, B_1 \sub X$ with disjoint interiors. -\item A field $r \in \cC(X \setmin (B_0 \cup B_1); c_0, c_1)$ -(where $c_i \in \cC(\bd B_i)$). -\item Local relation fields $u_i \in U(B_i; c_i)$, $i=1,2$. -\end{itemize} -We also identify $(B_0, B_1, u_0, u_1, r)$ with $-(B_1, B_0, u_1, u_0, r)$; -reversing the order of the blobs changes the sign. -Define $\bd(B_0, B_1, u_0, u_1, r) = -(B_1, u_1, u_0\bullet r) - (B_0, u_0, u_1\bullet r) \in \bc_1(X)$. -In other words, the boundary of a disjoint 2-blob diagram -is the sum (with alternating signs) -of the two ways of erasing one of the blobs. -It's easy to check that $\bd^2 = 0$. - -A nested 2-blob diagram consists of -\begin{itemize} -\item A pair of nested balls (blobs) $B_0 \sub B_1 \sub X$. -\item A field $r \in \cC(X \setmin B_0; c_0)$ -(for some $c_0 \in \cC(\bd B_0)$), which is cuttable along $\bd B_1$. -\item A local relation field $u_0 \in U(B_0; c_0)$. -\end{itemize} -Let $r = r_1 \bullet r'$, where $r_1 \in \cC(B_1 \setmin B_0; c_0, c_1)$ -(for some $c_1 \in \cC(B_1)$) and -$r' \in \cC(X \setmin B_1; c_1)$. -Define $\bd(B_0, B_1, u_0, r) = (B_1, u_0\bullet r_1, r') - (B_0, u_0, r)$. -Note that the requirement that -local relations are an ideal with respect to gluing guarantees that $u_0\bullet r_1 \in U(B_1)$. -As in the disjoint 2-blob case, the boundary of a nested 2-blob is the alternating -sum of the two ways of erasing one of the blobs. -If we erase the inner blob, the outer blob inherits the label $u_0\bullet r_1$. -It is again easy to check that $\bd^2 = 0$. - -\nn{should draw figures for 1, 2 and $k$-blob diagrams} - -As with the 1-blob diagrams, in order to get the linear structure correct it is better to define -(officially) -\begin{eqnarray*} - \bc_2(X) & \deq & - \left( - \bigoplus_{B_0, B_1 \text{disjoint}} \bigoplus_{c_0, c_1} - U(B_0; c_0) \otimes U(B_1; c_1) \otimes \lf(X\setmin (B_0\cup B_1); c_0, c_1) - \right) \\ - && \bigoplus \left( - \bigoplus_{B_0 \subset B_1} \bigoplus_{c_0} - U(B_0; c_0) \otimes \lf(X\setmin B_0; c_0) - \right) . -\end{eqnarray*} -The final $\lf(X\setmin B_0; c_0)$ above really means fields cuttable along $\bd B_1$, -but we didn't feel like introducing a notation for that. -For the disjoint blobs, reversing the ordering of $B_0$ and $B_1$ introduces a minus sign -(rather than a new, linearly independent 2-blob diagram). - -Now for the general case. -A $k$-blob diagram consists of -\begin{itemize} -\item A collection of blobs $B_i \sub X$, $i = 0, \ldots, k-1$. -For each $i$ and $j$, we require that either $B_i$ and $B_j$have disjoint interiors or -$B_i \sub B_j$ or $B_j \sub B_i$. -(The case $B_i = B_j$ is allowed. -If $B_i \sub B_j$ the boundaries of $B_i$ and $B_j$ are allowed to intersect.) -If a blob has no other blobs strictly contained in it, we call it a twig blob. -\item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$. -(These are implied by the data in the next bullets, so we usually -suppress them from the notation.) -$c_i$ and $c_j$ must have identical restrictions to $\bd B_i \cap \bd B_j$ -if the latter space is not empty. -\item A field $r \in \cC(X \setmin B^t; c^t)$, -where $B^t$ is the union of all the twig blobs and $c^t \in \cC(\bd B^t)$ -is determined by the $c_i$'s. -$r$ is required to be cuttable along the boundaries of all blobs, twigs or not. -\item For each twig blob $B_j$ a local relation field $u_j \in U(B_j; c_j)$, -where $c_j$ is the restriction of $c^t$ to $\bd B_j$. -If $B_i = B_j$ then $u_i = u_j$. -\end{itemize} - -If two blob diagrams $D_1$ and $D_2$ -differ only by a reordering of the blobs, then we identify -$D_1 = \pm D_2$, where the sign is the sign of the permutation relating $D_1$ and $D_2$. +\input{text/definitions} -$\bc_k(X)$ is, roughly, all finite linear combinations of $k$-blob diagrams. -As before, the official definition is in terms of direct sums -of tensor products: -\[ - \bc_k(X) \deq \bigoplus_{\overline{B}} \bigoplus_{\overline{c}} - \left( \otimes_j U(B_j; c_j)\right) \otimes \lf(X \setmin B^t; c^t) . -\] -Here $\overline{B}$ runs over all configurations of blobs, satisfying the conditions above. -$\overline{c}$ runs over all boundary conditions, again as described above. -$j$ runs over all indices of twig blobs. The final $\lf(X \setmin B^t; c^t)$ must be interpreted as fields which are cuttable along all of the blobs in $\overline{B}$. - -The boundary map $\bd : \bc_k(X) \to \bc_{k-1}(X)$ is defined as follows. -Let $b = (\{B_i\}, \{u_j\}, r)$ be a $k$-blob diagram. -Let $E_j(b)$ denote the result of erasing the $j$-th blob. -If $B_j$ is not a twig blob, this involves only decrementing -the indices of blobs $B_{j+1},\ldots,B_{k-1}$. -If $B_j$ is a twig blob, we have to assign new local relation labels -if removing $B_j$ creates new twig blobs. -If $B_l$ becomes a twig after removing $B_j$, then set $u_l = u_j\bullet r_l$, -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). -} -The $(-1)^j$ factors imply that the terms of $\bd^2(b)$ all cancel. -Thus we have a chain complex. - -\nn{?? say something about the ``shape" of tree? (incl = cone, disj = product)} - -\nn{?? remark about dendroidal sets} - - - -\section{Basic properties of the blob complex} -\label{sec:basic-properties} - -\begin{prop} \label{disjunion} -There is a natural isomorphism $\bc_*(X \du Y) \cong \bc_*(X) \otimes \bc_*(Y)$. -\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 -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) -to one that puts $X$ blobs before $Y$ blobs in the ordering, and so determines -a pair of blob diagrams on $X$ and $Y$. -These two maps are compatible with our sign conventions. -The two maps are inverses of each other. -\nn{should probably say something about sign conventions for the differential -in a tensor product of chain complexes; ask Scott} -\end{proof} - -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 $s: H_0(\bc_*(B^n, c)) \to \bc_0(B^n; c)$ -of the quotient map -$p: \bc_0(B^n; c) \to H_0(\bc_*(B^n, c))$. -For example, this is always the case if you coefficient ring is a field. -Then -\begin{prop} \label{bcontract} -For all $c \in \cC(\bd B^n)$ the natural map $p: \bc_*(B^n, c) \to H_0(\bc_*(B^n, c))$ -is a chain homotopy equivalence -with inverse $s: H_0(\bc_*(B^n, c)) \to \bc_*(B^n; c)$. -Here we think of $H_0(\bc_*(B^n, c))$ as a 1-step complex concentrated in degree 0. -\end{prop} -\begin{proof} -By assumption $p\circ s = \id$, so all that remains is to find a degree 1 map -$h : \bc_*(B^n; c) \to \bc_*(B^n; c)$ such that $\bd h + h\bd = \id - s \circ p$. -For $i \ge 1$, define $h_i : \bc_i(B^n; c) \to \bc_{i+1}(B^n; c)$ by adding -an $(i{+}1)$-st blob equal to all of $B^n$. -In other words, add a new outermost blob which encloses all of the others. -Define $h_0 : \bc_0(B^n; c) \to \bc_1(B^n; c)$ by setting $h_0(x)$ equal to -the 1-blob with blob $B^n$ and label $x - s(p(x)) \in U(B^n; c)$. -\end{proof} +\input{text/basic_properties} -Note that if there is no splitting $s$, we can let $h_0 = 0$ and get a homotopy -equivalence to the 2-step complex $U(B^n; c) \to \cC(B^n; c)$. - -For fields based on $n$-categories, $H_0(\bc_*(B^n; c)) \cong \mor(c', c'')$, -where $(c', c'')$ is some (any) splitting of $c$ into domain and range. - -\medskip - -\nn{Maybe there is no longer a need to repeat the next couple of props here, since we also state them in the introduction. -But I think it's worth saying that the Diff actions will be enhanced later. -Maybe put that in the intro too.} - -As we noted above, -\begin{prop} -There is a natural isomorphism $H_0(\bc_*(X)) \cong A(X)$. -\qed -\end{prop} - - -\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 -(chain map) isomorphisms. -\qed -\end{prop} - -In particular, -\begin{prop} \label{diff0prop} -There is an action of $\Diff(X)$ on $\bc_*(X)$. -\qed -\end{prop} - -The above will be greatly strengthened in Section \ref{sec:evaluation}. - -\medskip - -For the next proposition we will temporarily restore $n$-manifold boundary -conditions to the notation. - -Let $X$ be an $n$-manifold, $\bd X = Y \cup (-Y) \cup Z$. -Gluing the two copies of $Y$ together yields an $n$-manifold $X\sgl$ -with boundary $Z\sgl$. -Given compatible fields (boundary conditions) $a$, $b$ and $c$ on $Y$, $-Y$ and $Z$, -we have the blob complex $\bc_*(X; a, b, c)$. -If $b = -a$ (the orientation reversal of $a$), then we can glue up blob diagrams on -$X$ to get blob diagrams on $X\sgl$: - -\begin{prop} -There is a natural chain map -\eq{ - \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 -($n{-}2$-dimensional) boundaries with $c$. -`Natural' means natural with respect to the actions of diffeomorphisms. -\qed -\end{prop} - -The above map is very far from being an isomorphism, even on homology. -This will be fixed in Section \ref{sec:gluing} below. - -\nn{Next para not need, since we already use bullet = gluing notation above(?)} - -An instance of gluing we will encounter frequently below is where $X = X_1 \du X_2$ -and $X\sgl = X_1 \cup_Y X_2$. -(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) . -} -Note that we have resumed our habit of omitting boundary labels from the notation. - - - - - -\section{Hochschild homology when $n=1$} -\label{sec:hochschild} \input{text/hochschild} - - - -\section{Action of $\CD{X}$} -\label{sec:evaluation} \input{text/evmap} - - -\input{text/ncat.tex} - -\input{text/A-infty.tex} - -\input{text/gluing.tex} - - - -\section{Commutative algebras as $n$-categories} - -\nn{this should probably not be a section by itself. i'm just trying to write down the outline -while it's still fresh in my mind.} - -If $C$ is a commutative algebra it -can (and will) also be thought of as an $n$-category whose $j$-morphisms are trivial for -$j0$, and -of course $\Sigma^0(S^1)$ is a point. -Thus the singular homology $H_i(\Sigma^\infty(S^1))$ has infinitely many generators for $i=0,1$ -and is zero for $i\ge 2$. -\nn{say something about $t$-degrees also matching up?} +\input{text/gluing} -By xxxx and \ref{ktcdprop}, -the cyclic homology of $k[t]$ is the $S^1$-equivariant homology of $\Sigma^\infty(S^1)$. -Up to homotopy, $S^1$ acts by $j$-fold rotation on $\Sigma^j(S^1) \simeq S^1/j$. -If $k = \z$, $\Sigma^j(S^1)$ contributes the homology of an infinite lens space: $\z$ in degree -0, $\z/j \z$ in odd degrees, and 0 in positive even degrees. -The point $\Sigma^0(S^1)$ contributes the homology of $BS^1$ which is $\z$ in even -degrees and 0 in odd degrees. -This agrees with the calculation in \nn{Loday, 3.1.7}. - -\medskip - -Next we consider the case $C = k[t_1, \ldots, t_m]$, commutative polynomials in $m$ variables. -Let $\Sigma_m^\infty(M)$ be the $m$-colored infinite symmetric power of $M$, that is, configurations -of points on $M$ which can have any of $m$ distinct colors but are otherwise indistinguishable. -The components of $\Sigma_m^\infty(M)$ are indexed by $m$-tuples of natural numbers -corresponding to the number of points of each color of a configuration. -A proof similar to that of \ref{sympowerprop} shows that - -\begin{prop} -$\bc_*(M, k[t_1, \ldots, t_m])$ is homotopy equivalent to $C_*(\Sigma_m^\infty(M), k)$. -\end{prop} - -According to \nn{Loday, 3.2.2}, -\[ - HH_n(k[t_1, \ldots, t_m]) \cong \Lambda^n(k^m) \otimes k[t_1, \ldots, t_m] . -\] -Let us check that this is also the singular homology of $\Sigma_m^\infty(S^1)$. -We will content ourselves with the case $k = \z$. -One can define a flow on $\Sigma_m^\infty(S^1)$ where points of the same color repel each other and points of different colors do not interact. -This shows that a component $X$ of $\Sigma_m^\infty(S^1)$ is homotopy equivalent -to the torus $(S^1)^l$, where $l$ is the number of non-zero entries in the $m$-tuple -corresponding to $X$. -The homology calculation we desire follows easily from this. - -\nn{say something about cyclic homology in this case? probably not necessary.} - -\medskip - -Next we consider the case $C = k[t]/t^l$ --- polynomials in $t$ with $t^l = 0$. -Define $\Delta_l \sub \Sigma^\infty(M)$ to be those configurations of points with $l$ or -more points coinciding. - -\begin{prop} -$\bc_*(M, k[t]/t^l)$ is homotopy equivalent to $C_*(\Sigma^\infty(M), \Delta_l, k)$ -(relative singular chains with coefficients in $k$). -\end{prop} - -\begin{proof} -\nn{...} -\end{proof} - -\nn{...} - - +\input{text/comm_alg} \appendix -\input{text/famodiff.tex} - -\section{Comparing definitions of $A_\infty$ algebras} -\label{sec:comparing-A-infty} -In this section, we make contact between the usual definition of an $A_\infty$ algebra and our definition of a topological $A_\infty$ algebra, from Definition \ref{defn:topological-Ainfty-category}. - -We begin be restricting the data of a topological $A_\infty$ algebra to the standard interval $[0,1]$, which we can alternatively characterise as: -\begin{defn} -A \emph{topological $A_\infty$ category on $[0,1]$} $\cC$ has a set of objects $\Obj(\cC)$, and for each $a,b \in \Obj(\cC)$, a chain complex $\cC_{a,b}$, along with -\begin{itemize} -\item an action of the operad of $\Obj(\cC)$-labeled cell decompositions -\item and a compatible action of $\CD{[0,1]}$. -\end{itemize} -\end{defn} -Here the operad of cell decompositions of $[0,1]$ has operations indexed by a finite set of points $0 < x_1< \cdots < x_k < 1$, cutting $[0,1]$ into subintervals. An $X$-labeled cell decomposition labels $\{0, x_1, \ldots, x_k, 1\}$ by $X$. Given two cell decompositions $\cJ^{(1)}$ and $\cJ^{(2)}$, and an index $m$, we can compose them to form a new cell decomposition $\cJ^{(1)} \circ_m \cJ^{(2)}$ by inserting the points of $\cJ^{(2)}$ linearly into the $m$-th interval of $\cJ^{(1)}$. In the $X$-labeled case, we insist that the appropriate labels match up. Saying we have an action of this operad means that for each labeled cell decomposition $0 < x_1< \cdots < x_k < 1$, $a_0, \ldots, a_{k+1} \subset \Obj(\cC)$, there is a chain map $$\cC_{a_0,a_1} \tensor \cdots \tensor \cC_{a_k,a_{k+1}} \to \cC(a_0,a_{k+1})$$ and these chain maps compose exactly as the cell decompositions. -An action of $\CD{[0,1]}$ is compatible with an action of the cell decomposition operad if given a decomposition $\pi$, and a family of diffeomorphisms $f \in \CD{[0,1]}$ which is supported on the subintervals determined by $\pi$, then the two possible operations (glue intervals together, then apply the diffeomorphisms, or apply the diffeormorphisms separately to the subintervals, then glue) commute (as usual, up to a weakly unique homotopy). - -Translating between these definitions is straightforward. To restrict to the standard interval, define $\cC_{a,b} = \cC([0,1];a,b)$. Given a cell decomposition $0 < x_1< \cdots < x_k < 1$, we use the map (suppressing labels) -$$\cC([0,1])^{\tensor k+1} \to \cC([0,x_1]) \tensor \cdots \tensor \cC[x_k,1] \to \cC([0,1])$$ -where the factors of the first map are induced by the linear isometries $[0,1] \to [x_i, x_{i+1}]$, and the second map is just gluing. The action of $\CD{[0,1]}$ carries across, and is automatically compatible. Going the other way, we just declare $\cC(J;a,b) = \cC_{a,b}$, pick a diffeomorphism $\phi_J : J \isoto [0,1]$ for every interval $J$, define the gluing map $\cC(J_1) \tensor \cC(J_2) \to \cC(J_1 \cup J_2)$ by the first applying the cell decomposition map for $0 < \frac{1}{2} < 1$, then the self-diffeomorphism of $[0,1]$ given by $\frac{1}{2} (\phi_{J_1} \cup (1+ \phi_{J_2})) \circ \phi_{J_1 \cup J_2}^{-1}$. You can readily check that this gluing map is associative on the nose. \todo{really?} - -%First recall the \emph{coloured little intervals operad}. Given a set of labels $\cL$, the operations are indexed by \emph{decompositions of the interval}, each of which is a collection of disjoint subintervals $\{(a_i,b_i)\}_{i=1}^k$ of $[0,1]$, along with a labeling of the complementary regions by $\cL$, $\{l_0, \ldots, l_k\}$. Given two decompositions $\cJ^{(1)}$ and $\cJ^{(2)}$, and an index $m$ such that $l^{(1)}_{m-1} = l^{(2)}_0$ and $l^{(1)}_{m} = l^{(2)}_{k^{(2)}}$, we can form a new decomposition by inserting the intervals of $\cJ^{(2)}$ linearly inside the $m$-th interval of $\cJ^{(1)}$. We call the resulting decomposition $\cJ^{(1)} \circ_m \cJ^{(2)}$. - -%\begin{defn} -%A \emph{topological $A_\infty$ category} $\cC$ has a set of objects $\Obj(\cC)$ and for each $a,b \in \Obj(\cC)$ a chain complex $\cC_{a,b}$, along with a compatible `composition map' and an `action of families of diffeomorphisms'. - -%A \emph{composition map} $f$ is a family of chain maps, one for each decomposition of the interval, $f_\cJ : A^{\tensor k} \to A$, making $\cC$ into a category over the coloured little intervals operad, with labels $\cL = \Obj(\cC)$. Thus the chain maps satisfy the identity -%\begin{equation*} -%f_{\cJ^{(1)} \circ_m \cJ^{(2)}} = f_{\cJ^{(1)}} \circ (\id^{\tensor m-1} \tensor f_{\cJ^{(2)}} \tensor \id^{\tensor k^{(1)} - m}). -%\end{equation*} +\input{text/famodiff} -%An \emph{action of families of diffeomorphisms} is a chain map $ev: \CD{[0,1]} \tensor A \to A$, such that -%\begin{enumerate} -%\item The diagram -%\begin{equation*} -%\xymatrix{ -%\CD{[0,1]} \tensor \CD{[0,1]} \tensor A \ar[r]^{\id \tensor ev} \ar[d]^{\circ \tensor \id} & \CD{[0,1]} \tensor A \ar[d]^{ev} \\ -%\CD{[0,1]} \tensor A \ar[r]^{ev} & A -%} -%\end{equation*} -%commutes up to weakly unique homotopy. -%\item If $\phi \in \Diff([0,1])$ and $\cJ$ is a decomposition of the interval, we obtain a new decomposition $\phi(\cJ)$ and a collection $\phi_m \in \Diff([0,1])$ of diffeomorphisms obtained by taking the restrictions $\restrict{\phi}{[a_m,b_m]} : [a_m,b_m] \to [\phi(a_m),\phi(b_m)]$ and pre- and post-composing these with the linear diffeomorphisms $[0,1] \to [a_m,b_m]$ and $[\phi(a_m),\phi(b_m)] \to [0,1]$. We require that -%\begin{equation*} -%\phi(f_\cJ(a_1, \cdots, a_k)) = f_{\phi(\cJ)}(\phi_1(a_1), \cdots, \phi_k(a_k)). -%\end{equation*} -%\end{enumerate} -%\end{defn} - -From a topological $A_\infty$ category on $[0,1]$ $\cC$ we can produce a `conventional' $A_\infty$ category $(A, \{m_k\})$ as defined in, for example, \cite{MR1854636}. We'll just describe the algebra case (that is, a category with only one object), as the modifications required to deal with multiple objects are trivial. Define $A = \cC$ as a chain complex (so $m_1 = d$). Define $m_2 : A\tensor A \to A$ by $f_{\{(0,\frac{1}{2}),(\frac{1}{2},1)\}}$. To define $m_3$, we begin by taking the one parameter family $\phi_3$ of diffeomorphisms of $[0,1]$ that interpolates linearly between the identity and the piecewise linear diffeomorphism taking $\frac{1}{4}$ to $\frac{1}{2}$ and $\frac{1}{2}$ to $\frac{3}{4}$, and then define -\begin{equation*} -m_3(a,b,c) = ev(\phi_3, m_2(m_2(a,b), c)). -\end{equation*} +\input{text/misc_appendices} -It's then easy to calculate that -\begin{align*} -d(m_3(a,b,c)) & = ev(d \phi_3, m_2(m_2(a,b),c)) - ev(\phi_3 d m_2(m_2(a,b), c)) \\ - & = ev( \phi_3(1), m_2(m_2(a,b),c)) - ev(\phi_3(0), m_2 (m_2(a,b),c)) - \\ & \qquad - ev(\phi_3, m_2(m_2(da, b), c) + (-1)^{\deg a} m_2(m_2(a, db), c) + \\ & \qquad \quad + (-1)^{\deg a+\deg b} m_2(m_2(a, b), dc) \\ - & = m_2(a , m_2(b,c)) - m_2(m_2(a,b),c) - \\ & \qquad - m_3(da,b,c) + (-1)^{\deg a + 1} m_3(a,db,c) + \\ & \qquad \quad + (-1)^{\deg a + \deg b + 1} m_3(a,b,dc), \\ -\intertext{and thus that} -m_1 \circ m_3 & = m_2 \circ (\id \tensor m_2) - m_2 \circ (m_2 \tensor \id) - \\ & \qquad - m_3 \circ (m_1 \tensor \id \tensor \id) - m_3 \circ (\id \tensor m_1 \tensor \id) - m_3 \circ (\id \tensor \id \tensor m_1) -\end{align*} -as required (c.f. \cite[p. 6]{MR1854636}). -\todo{then the general case.} -We won't describe a reverse construction (producing a topological $A_\infty$ category from a `conventional' $A_\infty$ category), but we presume that this will be easy for the experts. - -\section{Morphisms and duals of topological $A_\infty$ modules} -\label{sec:A-infty-hom-and-duals}% - -\begin{defn} -If $\cM$ and $\cN$ are topological $A_\infty$ left modules over a topological $A_\infty$ category $\cC$, then a morphism $f: \cM \to \cN$ consists of a chain map $f:\cM(J,p;b) \to \cN(J,p;b)$ for each right marked interval $(J,p)$ with a boundary condition $b$, such that for each interval $J'$ the diagram -\begin{equation*} -\xymatrix{ -\cC(J';a,b) \tensor \cM(J,p;b) \ar[r]^{\text{gl}} \ar[d]^{\id \tensor f} & \cM(J' cup J,a) \ar[d]^f \\ -\cC(J';a,b) \tensor \cN(J,p;b) \ar[r]^{\text{gl}} & \cN(J' cup J,a) -} -\end{equation*} -commutes on the nose, and the diagram -\begin{equation*} -\xymatrix{ -\CD{(J,p) \to (J',p')} \tensor \cM(J,p;a) \ar[r]^{\text{ev}} \ar[d]^{\id \tensor f} & \cM(J',p';a) \ar[d]^f \\ -\CD{(J,p) \to (J',p')} \tensor \cN(J,p;a) \ar[r]^{\text{ev}} & \cN(J',p';a) \\ -} -\end{equation*} -commutes up to a weakly unique homotopy. -\end{defn} - -The variations required for right modules and bimodules should be obvious. - -\todo{duals} -\todo{ the functors $\hom_{\lmod{\cC}}\left(\cM \to -\right)$ and $\cM^* \tensor_{\cC} -$ from $\lmod{\cC}$ to $\Vect$ are naturally isomorphic} - - -\input{text/obsolete.tex} +\input{text/obsolete} % ---------------------------------------------------------------- %\newcommand{\urlprefix}{} @@ -967,9 +107,3 @@ % ---------------------------------------------------------------- - - -%Recall that for $n$-category picture fields there is an evaluation map -%$m: \bc_0(B^n; c, c') \to \mor(c, c')$. -%If we regard $\mor(c, c')$ as a complex concentrated in degree 0, then this becomes a chain -%map $m: \bc_*(B^n; c, c') \to \mor(c, c')$.