# HG changeset patch # User kevin@6e1638ff-ae45-0410-89bd-df963105f760 # Date 1267566721 0 # Node ID 1b3ebb7793c9e821ba143ec1a60df0649e883979 # Parent adc03f9d84223c388bd355abfdffa4916595cd4f ... diff -r adc03f9d8422 -r 1b3ebb7793c9 text/blobdef.tex --- a/text/blobdef.tex Tue Mar 02 20:07:31 2010 +0000 +++ b/text/blobdef.tex Tue Mar 02 21:52:01 2010 +0000 @@ -4,7 +4,10 @@ \label{sec:blob-definition} Let $X$ be an $n$-manifold. -Assume a fixed system of fields and local relations. +Let $\cC$ be a fixed system of fields (enriched over Vect) and local relations. +(If $\cC$ is not enriched over Vect, we can make it so by allowing finite +linear combinations of elements of $\cC(X; c)$, for fixed $c\in \cC(\bd X)$.) + In this section we will usually suppress boundary conditions on $X$ from the notation (e.g. write $\lf(X)$ instead of $\lf(X; c)$). @@ -22,11 +25,11 @@ We of course 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 $X$. +In other words, $\bc_0(X)$ is just the vector 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 +combinations of 1-blob diagrams, where a 1-blob diagram consists of \begin{itemize} \item An embedded closed ball (``blob") $B \sub X$. \item A field $r \in \cC(X \setmin B; c)$ @@ -35,7 +38,7 @@ (same $c$ as previous bullet). \end{itemize} (See Figure \ref{blob1diagram}.) -\begin{figure}[!ht]\begin{equation*} +\begin{figure}[t]\begin{equation*} \mathfig{.9}{definition/single-blob} \end{equation*}\caption{A 1-blob diagram.}\label{blob1diagram}\end{figure} In order to get the linear structure correct, we (officially) define @@ -50,8 +53,7 @@ \[ (B, u, r) \mapsto u\bullet r, \] -where $u\bullet r$ denotes the linear -combination of fields on $X$ obtained by gluing $u$ to $r$. +where $u\bullet r$ denotes the field 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 $u$). @@ -59,7 +61,7 @@ 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 +$\bc_2(X)$ is, roughly, the space of all relations (redundancies, syzygies) 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. @@ -72,7 +74,7 @@ \item Local relation fields $u_i \in U(B_i; c_i)$, $i=1,2$. \nn{We're inconsistent with the indexes -- are they 0,1 or 1,2? I'd prefer 1,2.} \end{itemize} (See Figure \ref{blob2ddiagram}.) -\begin{figure}[!ht]\begin{equation*} +\begin{figure}[t]\begin{equation*} \mathfig{.9}{definition/disjoint-blobs} \end{equation*}\caption{A disjoint 2-blob diagram.}\label{blob2ddiagram}\end{figure} We also identify $(B_0, B_1, u_0, u_1, r)$ with $-(B_1, B_0, u_1, u_0, r)$; @@ -92,7 +94,7 @@ \item A local relation field $u_0 \in U(B_0; c_0)$. \end{itemize} (See Figure \ref{blob2ndiagram}.) -\begin{figure}[!ht]\begin{equation*} +\begin{figure}[t]\begin{equation*} \mathfig{.9}{definition/nested-blobs} \end{equation*}\caption{A nested 2-blob diagram.}\label{blob2ndiagram}\end{figure} Let $r = r_1 \bullet r'$, where $r_1 \in \cC(B_1 \setmin B_0; c_0, c_1)$ @@ -128,7 +130,7 @@ 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 +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.) @@ -147,7 +149,7 @@ If $B_i = B_j$ then $u_i = u_j$. \end{itemize} (See Figure \ref{blobkdiagram}.) -\begin{figure}[!ht]\begin{equation*} +\begin{figure}[t]\begin{equation*} \mathfig{.9}{definition/k-blobs} \end{equation*}\caption{A $k$-blob diagram.}\label{blobkdiagram}\end{figure} @@ -166,7 +168,11 @@ $\overline{c}$ 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 splittable along all of the blobs in $\overline{B}$. -The boundary map $\bd : \bc_k(X) \to \bc_{k-1}(X)$ is defined as follows. +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 @@ -182,8 +188,20 @@ 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?; probably not} +We note that blob diagrams in $X$ have a structure similar to that of a simplicial set, +but with simplices replaced by a more general class of combinatorial shapes. +Let $P$ be the minimal set of (isomorphisms classes of) polyhedra which is closed under products +and cones, and which contains the point. +We can associate an element $p(b)$ of $P$ to each blob diagram $b$ +(equivalently, to each rooted tree) according to the following rules: +\begin{itemize} +\item $p(\emptyset) = pt$, where $\emptyset$ denotes a 0-blob diagram or empty tree; +\item $p(a \du b) = p(a) \times p(b)$, where $a \du b$ denotes the distant (non-overlapping) union of two blob diagrams (equivalently, join two trees at the roots); and +\item $p(\bar{b}) = \kone(p(b))$, where $\bar{b}$ is obtained from $b$ by adding an outer blob which encloses all the others. +\end{itemize} +(This correspondence works best if we thing of each twig label $u_i$ as being a difference of +two fields.) +For example, a diagram of $k$ strictly nested blobs corresponds to a $k$-simplex, while +a diagram of $k$ disjoint blobs corresponds to a $k$-cube. diff -r adc03f9d8422 -r 1b3ebb7793c9 text/kw_macros.tex --- a/text/kw_macros.tex Tue Mar 02 20:07:31 2010 +0000 +++ b/text/kw_macros.tex Tue Mar 02 21:52:01 2010 +0000 @@ -21,7 +21,8 @@ \def\op{^\mathrm{op}} \def\deq{\stackrel{\mathrm{def}}{=}} \def\pd#1#2{\frac{\partial #1}{\partial #2}} -\def\lf{\overline{\cC}} +%\def\lf{\overline{\cC}} +\def\lf{\cC} \def\ot{\otimes} \def\vphi{\varphi} \def\inv{^{-1}} @@ -58,6 +59,7 @@ \def\declaremathop#1{\expandafter\DeclareMathOperator\csname #1\endcsname{#1}} \applytolist{declaremathop}{pr}{im}{gl}{ev}{coinv}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{Tet}{cat}{Maps}{Diff}{Homeo}{sign}{supp}{Nbd}{res}; +\DeclareMathOperator{\kone}{cone} %%%%%% end excerpt