author | Scott Morrison <scott@tqft.net> |
Fri, 16 Jul 2010 17:18:02 -0600 | |
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%!TEX root = ../blob1.tex |
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\section{The blob complex} |
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\label{sec:blob-definition} |
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Let $X$ be an $n$-manifold. |
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Let $\cC$ be a fixed system of fields and local relations. |
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We'll assume it is enriched over \textbf{Vect}, and if it is not we can make it so by allowing finite |
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linear combinations of elements of $\cC(X; c)$, for fixed $c\in \cC(\bd X)$. |
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In this section we will usually suppress boundary conditions on $X$ from the notation, e.g. by writing $\lf(X)$ instead of $\lf(X; c)$. |
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We want to replace the quotient |
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\[ |
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A(X) \deq \lf(X) / U(X) |
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\] |
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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 $\bc_k(X)$. \todo{create a numbered definition for the general case} |
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We of course define $\bc_0(X) = \lf(X)$. |
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(If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \lf(X; c)$. |
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We'll omit this sort of detail in the rest of this section.) |
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In other words, $\bc_0(X)$ is just the vector space of all fields on $X$. |
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We want the vector space $\bc_1(X)$ to capture `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 An embedded closed ball (``blob") $B \sub X$. |
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\item A boundary condition $c \in \cC(\bdy B) = \cC(\bd(X \setmin B))$. |
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\item A field $r \in \cC(X \setmin B; c)$. |
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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, the actual definition is |
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\[ |
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\bc_1(X) \deq \bigoplus_B \bigoplus_c U(B; c) \otimes \lf(X \setmin B; c) . |
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\] |
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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 $c \in \cC(\bd B)$. |
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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{thm} |
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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{thm} |
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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 $r \in \cC(X \setmin (B_1 \cup B_2); c_1, c_2)$ |
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(where $c_i \in \cC(\bd B_i)$). |
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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 $r' \in \cC(B_2 \setminus B_1; c_1, c_2)$ |
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(for some $c_1 \in \cC(\bdy B_1)$ and $c_2 \in \cC(\bdy B_2)$). |
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\item A field $r \in \cC(X \setminus B_2; c_2)$. |
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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 \lf(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 \lf(B_2 \setmin B_1; c_1) \tensor \cC(X \setminus B_2; c_2) |
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\right) . |
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\end{eqnarray*} |
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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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\noop{ |
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\nn{Hmm, I think we should be doing this for nested blobs too -- |
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we shouldn't force the linear indexing of the blobs to have anything to do with |
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the partial ordering by inclusion -- this is what happens below} |
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\nn{KW: I think adding that detail would only add distracting clutter, and the statement is true as written (in the sense that it yields a vector space isomorphic to the general def below} |
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} |
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Before describing the general case we should say more precisely what we mean by |
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disjoint and nested blobs. |
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Disjoint will mean disjoint interiors. |
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Nested blobs are allowed to coincide, or to have overlapping boundaries. |
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Blob are allowed to intersect $\bd X$. |
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However, we require of any collection of blobs $B_1,\ldots,B_k \subseteq X$ that |
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$X$ is decomposable along the union of the boundaries of the blobs. |
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\nn{need to say more here. we want to be able to glue blob diagrams, but avoid pathological |
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behavior} |
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\nn{need to allow the case where $B\to X$ is not an embedding |
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on $\bd B$. this is because any blob diagram on $X_{cut}$ should give rise to one on $X_{gl}$ |
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and blobs are allowed to meet $\bd X$. |
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Also, the complement of the blobs (and regions between nested blobs) might not be manifolds.} |
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Now for the general case. |
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A $k$-blob diagram consists of |
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\begin{itemize} |
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\item A collection of blobs $B_i \sub X$, $i = 1, \ldots, k$. |
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For each $i$ and $j$, we require that either $B_i$ and $B_j$ have disjoint interiors or |
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$B_i \sub B_j$ or $B_j \sub B_i$. |
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(The case $B_i = B_j$ is allowed. |
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If $B_i \sub B_j$ the boundaries of $B_i$ and $B_j$ are allowed to intersect.) |
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If a blob has no other blobs strictly contained in it, we call it a twig blob. |
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\item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$. |
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(These are implied by the data in the next bullets, so we usually |
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suppress them from the notation.) |
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The fields $c_i$ and $c_j$ must have identical restrictions to $\bd B_i \cap \bd B_j$ |
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if the latter space is not empty. |
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\item A field $r \in \cC(X \setmin B^t; c^t)$, |
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where $B^t$ is the union of all the twig blobs and $c^t \in \cC(\bd B^t)$ |
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is determined by the $c_i$'s. |
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$r$ is required to be splittable along the boundaries of all blobs, twigs or not. (This is equivalent to asking for a field on of the components of $X \setmin B^t$.) |
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\item For each twig blob $B_j$ a local relation field $u_j \in U(B_j; c_j)$, |
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where $c_j$ is the restriction of $c^t$ to $\bd B_j$. |
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If $B_i = B_j$ then $u_i = u_j$. |
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\end{itemize} |
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(See Figure \ref{blobkdiagram}.) |
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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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If two blob diagrams $D_1$ and $D_2$ |
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differ only by a reordering of the blobs, then we identify |
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$D_1 = \pm D_2$, where the sign is the sign of the permutation relating $D_1$ and $D_2$. |
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Roughly, then, $\bc_k(X)$ is all finite linear combinations of $k$-blob diagrams. |
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As before, the official definition is in terms of direct sums |
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of tensor products: |
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\[ |
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\bc_k(X) \deq \bigoplus_{\overline{B}} \bigoplus_{\overline{c}} |
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\left( \bigotimes_j U(B_j; c_j)\right) \otimes \lf(X \setmin B^t; c^t) . |
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\] |
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Here $\overline{B}$ runs over all configurations of blobs, satisfying the conditions above. |
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The index $\overline{c}$ runs over all boundary conditions, again as described above and $j$ runs over all indices of twig blobs. |
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The final $\lf(X \setmin B^t; c^t)$ must be interpreted as fields which are splittable along all of the blobs in $\overline{B}$. |
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The boundary map |
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\[ |
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\bd : \bc_k(X) \to \bc_{k-1}(X) |
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\] |
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is defined as follows. |
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Let $b = (\{B_i\}, \{u_j\}, r)$ be a $k$-blob diagram. |
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Let $E_j(b)$ denote the result of erasing the $j$-th blob. |
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If $B_j$ is not a twig blob, this involves only decrementing |
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the indices of blobs $B_{j+1},\ldots,B_{k}$. |
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If $B_j$ is a twig blob, we have to assign new local relation labels |
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if removing $B_j$ creates new twig blobs. \todo{Have to say what happens when no new twig blobs are created} |
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If $B_l$ becomes a twig after removing $B_j$, then set $u_l = u_j\bullet r_l$, |
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where $r_l$ is the restriction of $r$ to $B_l \setmin B_j$. |
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Finally, define |
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\eq{ |
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\bd(b) = \sum_{j=1}^{k} (-1)^{j+1} E_j(b). |
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} |
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The $(-1)^{j+1}$ factors imply that the terms of $\bd^2(b)$ all cancel. |
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Thus we have a chain complex. |
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Note that Property \ref{property:functoriality}, that the blob complex is functorial with respect to homeomorphisms, 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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We define the {\it support} of a blob diagram $b$, $\supp(b) \sub X$, |
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to be the union of the blobs of $b$. |
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For $y \in \bc_*(X)$ with $y = \sum c_i b_i$ ($c_i$ a non-zero number, $b_i$ a blob diagram), |
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we define $\supp(y) \deq \bigcup_i \supp(b_i)$. |
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We note that blob diagrams in $X$ have a structure similar to that of a simplicial set, |
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but with simplices replaced by a more general class of combinatorial shapes. |
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Let $P$ be the minimal set of (isomorphisms classes of) polyhedra which is closed under products |
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and cones, and which contains the point. |
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We can associate an element $p(b)$ of $P$ to each blob diagram $b$ |
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(equivalently, to each rooted tree) according to the following rules: |
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\begin{itemize} |
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\item $p(\emptyset) = pt$, where $\emptyset$ denotes a 0-blob diagram or empty tree; |
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\item $p(a \du b) = p(a) \times p(b)$, where $a \du b$ denotes the distant (non-overlapping) union |
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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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\end{itemize} |
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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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(This correspondence works best if we think of each twig label $u_i$ as having the form |
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$x - s(e(x))$, where $x$ is an arbitrary field on $B_i$, $e: \cC(B_i) \to C$ is the evaluation map, |
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and $s:C \to \cC(B_i)$ is some fixed section of $e$. \todo{This parenthetical remark mysteriously specialises to the category case}) |
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