author | Kevin Walker <kevin@canyon23.net> |
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%!TEX root = ../blob1.tex |
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\section{The blob complex} |
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\subsection{Definitions} |
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\label{sec:blob-definition} |
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Let $X$ be an $n$-manifold. |
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Let $(\cF,U)$ be a fixed system of fields and local relations. |
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We'll assume it is enriched over \textbf{Vect}; |
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if it is not we can make it so by allowing finite |
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linear combinations of elements of $\cF(X; c)$, for fixed $c\in \cF(\bd X)$. |
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%In this section we will usually suppress boundary conditions on $X$ from the notation, e.g. by writing $\cF(X)$ instead of $\cF(X; c)$. |
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We want to replace the quotient |
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\[ |
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A(X) \deq \cF(X) / U(X) |
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\] |
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of Definition \ref{defn:TQFT-invariant} with a resolution |
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\[ |
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\cdots \to \bc_2(X) \to \bc_1(X) \to \bc_0(X) . |
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\] |
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We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case $\bc_k(X)$. |
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In fact, on the first pass we will intentionally describe the definition in a misleadingly simple way, |
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then explain the technical difficulties, and finally give a cumbersome but complete definition in |
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Definition \ref{defn:blobs}. |
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If (we don't recommend it) you want to keep track of the ways in which |
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this initial description is misleading, or you're reading through a second time to understand the |
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technical difficulties, keep note that later we will give precise meanings to ``a ball in $X$'', |
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``nested'' and ``disjoint'', that are not quite the intuitive ones. |
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Moreover some of the pieces |
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into which we cut manifolds below are not themselves manifolds, and it requires special attention |
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to define fields on these pieces. |
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We of course define $\bc_0(X) = \cF(X)$. |
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In other words, $\bc_0(X)$ is just the vector space of all fields on $X$. |
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(If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \cF(X; c)$ for $c \in \cF(\bdy X)$. |
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The blob complex $\bc_*(X; c)$ will depend on a fixed boundary condition $c\in \cF(\bdy X)$. |
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We'll omit such boundary conditions from the notation in the rest of this section.) |
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We want the vector space $\bc_1(X)$ to capture |
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``the space of all local relations that can be imposed on $\bc_0(X)$". |
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Thus we say a $1$-blob diagram consists of: |
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\begin{itemize} |
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\item A closed ball in $X$ (``blob") $B \sub X$. |
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\item A boundary condition $c \in \cF(\bdy B) = \cF(\bd(X \setmin B))$. |
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\item A field $r \in \cF(X \setmin B; c)$. |
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\item A local relation field $u \in U(B; c)$. |
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\end{itemize} |
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(See Figure \ref{blob1diagram}.) Since $c$ is implicitly determined by $u$ or $r$, we usually omit it from the notation. |
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\begin{figure}[t]\begin{equation*} |
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\mathfig{.6}{definition/single-blob} |
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\end{equation*}\caption{A 1-blob diagram.}\label{blob1diagram}\end{figure} |
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In order to get the linear structure correct, we define |
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\[ |
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\bc_1(X) \deq \bigoplus_B \bigoplus_c U(B; c) \otimes \cF(X \setmin B; c) . |
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\] |
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The first direct sum is indexed by all blobs $B\subset X$, and the second |
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by all boundary conditions $c \in \cF(\bd B)$. |
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Note that $\bc_1(X)$ is spanned by 1-blob diagrams $(B, u, r)$. |
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Define the boundary map $\bd : \bc_1(X) \to \bc_0(X)$ by |
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\[ |
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(B, u, r) \mapsto u\bullet r, |
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\] |
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where $u\bullet r$ denotes the field on $X$ obtained by gluing $u$ to $r$. |
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In other words $\bd : \bc_1(X) \to \bc_0(X)$ is given by |
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just erasing the blob from the picture |
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(but keeping the blob label $u$). |
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Note that directly from the definition we have |
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\begin{prop} |
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\label{thm:skein-modules} |
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The skein module $A(X)$ is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$. |
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\end{prop} |
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This also establishes the second |
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half of Property \ref{property:contractibility}. |
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Next, we want the vector space $\bc_2(X)$ to capture ``the space of all relations |
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(redundancies, syzygies) among the |
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local relations encoded in $\bc_1(X)$''. |
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A $2$-blob diagram comes in one of two types, disjoint and nested. |
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A disjoint 2-blob diagram consists of |
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\begin{itemize} |
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\item A pair of closed balls (blobs) $B_1, B_2 \sub X$ with disjoint interiors. |
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\item A field $r \in \cF(X \setmin (B_1 \cup B_2); c_1, c_2)$ |
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(where $c_i \in \cF(\bd B_i)$). |
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\item Local relation fields $u_i \in U(B_i; c_i)$, $i=1,2$. |
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\end{itemize} |
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(See Figure \ref{blob2ddiagram}.) |
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\begin{figure}[t]\begin{equation*} |
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\mathfig{.6}{definition/disjoint-blobs} |
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\end{equation*}\caption{A disjoint 2-blob diagram.}\label{blob2ddiagram}\end{figure} |
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We also identify $(B_1, B_2, u_1, u_2, r)$ with $-(B_2, B_1, u_2, u_1, r)$; |
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reversing the order of the blobs changes the sign. |
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Define $\bd(B_1, B_2, u_1, u_2, r) = |
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(B_2, u_2, u_1\bullet r) - (B_1, u_1, u_2\bullet r) \in \bc_1(X)$. |
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In other words, the boundary of a disjoint 2-blob diagram |
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is the sum (with alternating signs) |
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of the two ways of erasing one of the blobs. |
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It's easy to check that $\bd^2 = 0$. |
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A nested 2-blob diagram consists of |
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\begin{itemize} |
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\item A pair of nested balls (blobs) $B_1 \subseteq B_2 \subseteq X$. |
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\item A field $r' \in \cF(B_2 \setminus B_1; c_1, c_2)$ |
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(for some $c_1 \in \cF(\bdy B_1)$ and $c_2 \in \cF(\bdy B_2)$). |
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\item A field $r \in \cF(X \setminus B_2; c_2)$. |
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\item A local relation field $u \in U(B_1; c_1)$. |
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\end{itemize} |
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(See Figure \ref{blob2ndiagram}.) |
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\begin{figure}[t]\begin{equation*} |
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\mathfig{.6}{definition/nested-blobs} |
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\end{equation*}\caption{A nested 2-blob diagram.}\label{blob2ndiagram}\end{figure} |
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Define $\bd(B_1, B_2, u, r', r) = (B_2, u\bullet r', r) - (B_1, u, r' \bullet r)$. |
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As in the disjoint 2-blob case, the boundary of a nested 2-blob is the alternating |
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sum of the two ways of erasing one of the blobs. |
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When we erase the inner blob, the outer blob inherits the label $u\bullet r'$. |
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It is again easy to check that $\bd^2 = 0$. Note that the requirement that |
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local relations are an ideal with respect to gluing guarantees that $u\bullet r' \in U(B_2)$. |
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As with the $1$-blob diagrams, in order to get the linear structure correct the actual definition is |
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\begin{eqnarray*} |
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\bc_2(X) & \deq & |
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\left( |
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\bigoplus_{B_1, B_2\; \text{disjoint}} \bigoplus_{c_1, c_2} |
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U(B_1; c_1) \otimes U(B_2; c_2) \otimes \cF(X\setmin (B_1\cup B_2); c_1, c_2) |
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\right) \bigoplus \\ |
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&& \quad\quad \left( |
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\bigoplus_{B_1 \subset B_2} \bigoplus_{c_1, c_2} |
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U(B_1; c_1) \otimes \cF(B_2 \setmin B_1; c_1, c_2) \tensor \cF(X \setminus B_2; c_2) |
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\right) . |
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\end{eqnarray*} |
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% __ (already said this above) |
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%For the disjoint blobs, reversing the ordering of $B_1$ and $B_2$ introduces a minus sign |
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%(rather than a new, linearly independent, 2-blob diagram). |
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\medskip |
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Roughly, $\bc_k(X)$ is generated by configurations of $k$ blobs, pairwise disjoint or nested, |
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along with fields on all the components that the blobs divide $X$ into. |
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Blobs which have no other blobs inside are called `twig blobs', |
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and the fields on the twig blobs must be local relations. |
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The boundary is the alternating sum of erasing one of the blobs. |
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In order to describe this general case in full detail, we must give a more precise description of |
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which configurations of balls inside $X$ we permit. |
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These configurations are generated by two operations: |
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\begin{itemize} |
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\item For any (possibly empty) configuration of blobs on an $n$-ball $D$, we can add |
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$D$ itself as an outermost blob. |
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(This is used in the proof of Proposition \ref{bcontract}.) |
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\item If $X\sgl$ is obtained from $X$ by gluing, then any permissible configuration of blobs |
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on $X$ gives rise to a permissible configuration on $X\sgl$. |
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(This is necessary for Proposition \ref{blob-gluing}.) |
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\end{itemize} |
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Combining these two operations can give rise to configurations of blobs whose complement in $X$ is not |
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a manifold. |
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Thus we will need to be more careful when speaking of a field $r$ on the complement of the blobs. |
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\begin{example} \label{sin1x-example} |
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Consider the four subsets of $\Real^3$, |
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\begin{align*} |
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A & = [0,1] \times [0,1] \times [0,1] \\ |
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B & = [0,1] \times [-1,0] \times [0,1] \\ |
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C & = [-1,0] \times \setc{(y,z)}{e^{-1/z^2} \sin(1/z) \leq y \leq 1, z \in [0,1]} \\ |
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D & = [-1,0] \times \setc{(y,z)}{-1 \leq y \leq e^{-1/z^2} \sin(1/z), z \in [0,1]}. |
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\end{align*} |
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Here $A \cup B = [0,1] \times [-1,1] \times [0,1]$ and $C \cup D = [-1,0] \times [-1,1] \times [0,1]$. |
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Now, $\{A\}$ is a valid configuration of blobs in $A \cup B$, |
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and $\{D\}$ is a valid configuration of blobs in $C \cup D$, |
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so we must allow $\{A, D\}$ as a configuration of blobs in $[-1,1]^2 \times [0,1]$. |
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Note however that the complement is not a manifold. See Figure \ref{fig:blocks}. |
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\end{example} |
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\begin{figure}[t]\begin{equation*} |
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\mathfig{.4}{definition/blocks} |
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\end{equation*}\caption{The subsets $A$, $B$, $C$ and $D$ from Example \ref{sin1x-example}. The pair $\{A, D\}$ is a valid configuration of blobs, even though the complement is not a manifold.}\label{fig:blocks}\end{figure} |
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\begin{defn} |
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\label{defn:gluing-decomposition} |
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A \emph{gluing decomposition} of an $n$-manifold $X$ is a sequence of manifolds |
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$M_0 \to M_1 \to \cdots \to M_m = X$ such that each $M_k$ is obtained from $M_{k-1}$ |
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by gluing together some disjoint pair of homeomorphic $n{-}1$-manifolds in the boundary of $M_{k-1}$. |
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If, in addition, $M_0$ is a disjoint union of balls, we call it a \emph{ball decomposition}. |
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\end{defn} |
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Let $M_0 \to M_1 \to \cdots \to M_m = X$ be a gluing decomposition of $X$, |
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and let $M_0^0,\ldots,M_0^k$ be the connected components of $M_0$. |
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We say that a field |
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$a\in \cF(X)$ is splittable along the decomposition if $a$ is the image |
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under gluing and disjoint union of fields $a_i \in \cF(M_0^i)$, $0\le i\le k$. |
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Note that if $a$ is splittable in this sense then it makes sense to talk about the restriction of $a$ to any |
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component $M'_j$ of any $M_j$ of the decomposition. |
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In the example above, note that |
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\[ |
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A \sqcup B \sqcup C \sqcup D \to (A \cup B) \sqcup (C \cup D) \to A \cup B \cup C \cup D |
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\] |
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is a ball decomposition, but other sequences of gluings starting from $A \sqcup B \sqcup C \sqcup D$ |
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have intermediate steps which are not manifolds. |
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We'll now slightly restrict the possible configurations of blobs. |
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%%%%% oops -- I missed the similar discussion after the definition |
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%The basic idea is that each blob in a configuration |
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%is the image a ball, with embedded interior and possibly glued-up boundary; |
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%distinct blobs should either have disjoint interiors or be nested; |
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%and the entire configuration should be compatible with some gluing decomposition of $X$. |
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\begin{defn} |
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\label{defn:configuration} |
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A configuration of $k$ blobs in $X$ is an ordered collection of $k$ subsets $\{B_1, \ldots, B_k\}$ |
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of $X$ such that there exists a gluing decomposition $M_0 \to \cdots \to M_m = X$ of $X$ |
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with the property that |
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for each subset $B_i$ there is some $0 \leq l \leq m$ and some connected component $M_l'$ of |
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$M_l$ which is a ball, such that $B_i$ is the image of $M_l'$ in $X$. |
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We say that such a gluing decomposition |
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is \emph{compatible} with the configuration. |
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A blob $B_i$ is a twig blob if no other blob $B_j$ is a strict subset of it. |
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\end{defn} |
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In particular, this implies what we said about blobs above: |
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that for any two blobs in a configuration of blobs in $X$, |
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they either have disjoint interiors, or one blob is contained in the other. |
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We describe these as disjoint blobs and nested blobs. |
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Note that nested blobs may have boundaries that overlap, or indeed coincide. |
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Blobs may meet the boundary of $X$. |
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Further, note that blobs need not actually be embedded balls in $X$, since parts of the |
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boundary of the ball $M_l'$ may have been glued together. |
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Note that often the gluing decomposition for a configuration of blobs may just be the trivial one: |
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if the boundaries of all the blobs cut $X$ into pieces which are all manifolds, |
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we can just take $M_0$ to be these pieces, and $M_1 = X$. |
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In the initial informal definition of a $k$-blob diagram above, we allowed any |
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collection of $k$ balls which were pairwise disjoint or nested. |
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We now further require that the balls are a configuration in the sense of Definition \ref{defn:configuration}. |
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We also specified a local relation on each twig blob, and a field on the complement of the twig blobs; |
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this is unsatisfactory because that complement need not be a manifold. Thus, the official definitions are |
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\begin{defn} |
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\label{defn:blob-diagram} |
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A $k$-blob diagram on $X$ consists of |
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\begin{itemize} |
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\item a configuration $\{B_1, \ldots, B_k\}$ of $k$ blobs in $X$, |
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\item and a field $r \in \cF(X)$ which is splittable along some gluing decomposition compatible with that configuration, |
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\end{itemize} |
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such that |
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the restriction $u_i$ of $r$ to each twig blob $B_i$ lies in the subspace |
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$U(B_i) \subset \cF(B_i)$. |
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(See Figure \ref{blobkdiagram}.) |
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More precisely, each twig blob $B_i$ is the image of some ball $M_l'$ as above, |
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and it is really the restriction to $M_l'$ that must lie in the subspace $U(M_l')$. |
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\end{defn} |
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\begin{figure}[t]\begin{equation*} |
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\mathfig{.7}{definition/k-blobs} |
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\end{equation*}\caption{A $k$-blob diagram.}\label{blobkdiagram}\end{figure} |
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\begin{defn} |
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\label{defn:blobs} |
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The $k$-th vector space $\bc_k(X)$ of the \emph{blob complex} of $X$ is the direct sum over all |
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configurations of $k$ blobs in $X$ of the vector space of $k$-blob diagrams with that configuration, |
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modulo identifying the vector spaces for configurations that only differ by a permutation of the blobs |
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by the sign of that permutation. |
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The differential $\bc_k(X) \to \bc_{k-1}(X)$ is, as above, the signed sum of ways of |
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forgetting one blob from the configuration, preserving the field $r$: |
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\begin{equation*} |
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\bdy(\{B_1, \ldots B_k\}, r) = \sum_{i=1}^{k} (-1)^{i+1} (\{B_1, \ldots, \widehat{B_i}, \ldots, B_k\}, r) |
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\end{equation*} |
474 | 267 |
\end{defn} |
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We readily see that if a gluing decomposition is compatible with some configuration of blobs, |
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then it is also compatible with any configuration obtained by forgetting some blobs, |
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ensuring that the differential in fact lands in the space of $k{-}1$-blob diagrams. |
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A slight compensation to the complication of the official definition arising from attention |
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to splitting is that the differential now just preserves the entire field $r$ without |
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having to say anything about gluing together fields on smaller components. |
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474 | 275 |
Note that Property \ref{property:functoriality}, that the blob complex is functorial with respect to homeomorphisms, |
276 |
is immediately obvious from the definition. |
|
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A homeomorphism acts in an obvious way on blobs and on fields. |
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\begin{remark} \label{blobsset-remark} \rm |
216 | 280 |
We note that blob diagrams in $X$ have a structure similar to that of a simplicial set, |
281 |
but with simplices replaced by a more general class of combinatorial shapes. |
|
282 |
Let $P$ be the minimal set of (isomorphisms classes of) polyhedra which is closed under products |
|
283 |
and cones, and which contains the point. |
|
284 |
We can associate an element $p(b)$ of $P$ to each blob diagram $b$ |
|
285 |
(equivalently, to each rooted tree) according to the following rules: |
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\begin{itemize} |
|
287 |
\item $p(\emptyset) = pt$, where $\emptyset$ denotes a 0-blob diagram or empty tree; |
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342 | 288 |
\item $p(a \du b) = p(a) \times p(b)$, where $a \du b$ denotes the distant (non-overlapping) union |
289 |
of two blob diagrams (equivalently, join two trees at the roots); and |
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\item $p(\bar{b}) = \kone(p(b))$, where $\bar{b}$ is obtained from $b$ by adding an outer blob which |
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encloses all the others (equivalently, add a new edge to the root, with the new vertex becoming the root |
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of the new tree). |
216 | 293 |
\end{itemize} |
294 |
For example, a diagram of $k$ strictly nested blobs corresponds to a $k$-simplex, while |
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a diagram of $k$ disjoint blobs corresponds to a $k$-cube. |
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(When the fields come from an $n$-category, this correspondence works best if we think of each |
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twig label $u_i$ as having the form |
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$x - s(e(x))$, where $x$ is an arbitrary field on $B_i$, $e: \cF(B_i) \to C$ is the evaluation map, |
491 | 299 |
and $s:C \to \cF(B_i)$ is some fixed section of $e$.) |
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For lack of a better name, |
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we'll call elements of $P$ cone-product polyhedra, |
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and say that blob diagrams have the structure of a cone-product set (analogous to simplicial set). |
513 | 304 |
\end{remark} |
215 | 305 |