blob: comparison text/blobdef.tex

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-:270164061847
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 \end{eqnarray*}
 % __ (already said this above)
 %For the disjoint blobs, reversing the ordering of $B_1$ and $B_2$ introduces a minus sign
 %(rather than a new, linearly independent, 2-blob diagram).
+\medskip
+Roughly, $\bc_k(X)$ is generated by configurations of $k$ blobs, pairwise disjoint or nested.
-Before describing the general case, note that when we say blobs are disjoint, we will only mean that their interiors are disjoint. Nested blobs may have boundaries that overlap, or indeed may coincide.
+The boundary is the alternating sum of erasing one of the blobs.
+In order to describe this general case in full detail, we must give a more precise description of
+which configurations of balls inside $X$ we permit.
+These configurations are generated by two operations:
+\begin{itemize}
+\item For any (possibly empty) configuration of blobs on an $n$-ball $D$, we can add
+$D$ itself as an outermost blob.
+(This is used in the proof of Proposition \ref{bcontract}.)
+\item If $X'$ is obtained from $X$ by gluing, then any permissible configuration of blobs
+on $X$ gives rise to a permissible configuration on $X'$.
+(This is necessary for Proposition \ref{blob-gluing}.)
+\end{itemize}
+Combining these two operations can give rise to configurations of blobs whose complement in $X$ is not
+a manifold.
+Thus will need to be more careful when speaking of a field $r$ on the complement of the blobs.
+%In order to precisely state the general definition, we'll need a suitable notion of cutting up a manifold into balls.
+\begin{defn}
+\label{defn:gluing-decomposition}
+A \emph{gluing decomposition} of an $n$-manifold $X$ is a sequence of manifolds
+$M_0 \to M_1 \to \cdots \to M_m = X$ such that each $M_k$ is obtained from $M_{k-1}$
+by gluing together some disjoint pair of homeomorphic $n{-}1$-manifolds in the boundary of $M_{k-1}$.
+If, in addition, $M_0$ is a disjoint union of balls, we call it a \emph{ball decomposition}.
+\end{defn}
+Given a gluing decomposition $M_0 \to M_1 \to \cdots \to M_m = X$, we say that a field is splittable along it if it is the image of a field on $M_0$.
+By ``a ball in $X$'' we don't literally mean a submanifold homeomorphic to a ball, but rather the image of a map from the pair $(B^n, S^{n-1})$ into $X$, which is an embedding on the interior. The boundary of a ball in $X$ is the image of a locally embedded $n{-}1$-sphere. \todo{examples, e.g. balls which actually look like an annulus, but we remember the boundary}
+\nn{not all balls in $X$ can arise via gluing, but I suppose that's OK.}
+\nn{do we need this next def?}
+\begin{defn}
+\label{defn:ball-decomposition}
+A \emph{ball decomposition} of an $n$-manifold $X$ is a collection of balls in $X$, such that there exists some gluing decomposition $M_0  \to \cdots \to M_m = X$ so that the balls are the images of the components of $M_0$ in $X$.
+\end{defn}
+In particular, the union of all the balls in a ball decomposition comprises all of $X$. \todo{example}
+We'll now slightly restrict the possible configurations of blobs.
+\begin{defn}
+\label{defn:configuration}
+A configuration of $k$ blobs in $X$ is an ordered collection of $k$ balls in $X$ such that there is some gluing decomposition $M_0  \to \cdots \to M_m = X$ of $X$ and each of the balls is the image of some connected component of one of the $M_k$. Such a gluing decomposition is \emph{compatible} with the configuration.
+\end{defn}
+In particular, this implies what we said about blobs above:
+that for any two blobs in a configuration of blobs in $X$,
+they either have disjoint interiors, or one blob is contained in the other.
+We describe these as disjoint blobs and nested blobs.
+Note that nested blobs may have boundaries that overlap, or indeed coincide.
+Blobs may meet the boundary of $X$.
+% (already said above)
+%Note that the boundaries of a configuration of $k$ blobs may cut up the manifold $X$ into components which are not themselves manifolds. \todo{example: the components between the boundaries of the balls may be pathological}
+In the informal description above, in the definition of a $k$-blob diagram we asked for any collection of $k$ balls which were pairwise disjoint or nested. We now further insist that the balls are a configuration in the sense of Definition \ref{defn:configuration}. Also, we asked for a local relation on each twig blob, and a field on the complement of the twig blobs; this is unsatisfactory because that complement need not be a manifold. Thus, the official definition is
+\begin{defn}
+\label{defn:blob-diagram}
+A $k$-blob diagram on $X$ consists of
+\begin{itemize}
+\item a configuration of $k$ blobs in $X$,
+\item and a field $r \in \cC(X)$ which is splittable along some gluing decomposition compatible with that configuration,
+\end{itemize}
+such that
+the restriction of $r$ to each twig blob $B_i$ lies in the subspace $U(B_i) \subset \cC(B_i)$.
+\end{defn}
+and
+\begin{defn}
+\label{defn:blobs}
+The $k$-th vector space $\bc_k(X)$ of the \emph{blob complex} of $X$ is the direct sum of all configurations of $k$ blobs in $X$ of the vector space of $k$-blob diagrams with that configuration, modulo identifying the vector spaces for configurations that only differ by a permutation of the balls by the sign of that permutation. The differential $bc_k(X) \to bc_{k-1}(X)$ is, as above, the signed sum of ways of forgetting one ball from the configuration, preserving the field $r$.
+\end{defn}
+We readily see that if a gluing decomposition is compatible with some configuration of blobs, then it is also compatible with any configuration obtained by forgetting some blobs, ensuring that the differential in fact lands in the space of $k{-}1$-blob diagrams.
+A slight compensation to the complication of the official definition arising from attention to splitting is that the differential now just preserves the entire field $r$ without having to say anything about gluing together fields on smaller components.
+\nn{should merge this informal def with official one above}
+Before describing the general case, note that when we say blobs are disjoint,
+we will only mean that their interiors are disjoint.
+Nested blobs may have boundaries that overlap, or indeed may coincide.
 A $k$-blob diagram consists of
 \begin{itemize}
 \item A collection of blobs $B_i \sub X$, $i = 1, \ldots, k$.
 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 fields $c_i$ 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.
-The field $r$ 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 $X \setmin B^t$.)
+The field $r$ 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 $X \setmin B^t$.)
 \item For each twig blob $B_j$ a local relation field $u_j \in U(B_j; c_j)$.
 If $B_i = B_j$ then $u_i = u_j$.
 \end{itemize}
 (See Figure \ref{blobkdiagram}.)
 \begin{figure}[t]\begin{equation*}
 \[
 	\bc_k(X) \deq \bigoplus_{\overline{B}} \bigoplus_{\overline{c}}
 		\left( \bigotimes_j U(B_j; c_j)\right) \otimes \lf(X \setmin B^t; c^t) .
 \]
 Here $\overline{B}$ runs over all configurations of blobs, satisfying the conditions above.
-The index $\overline{c}$ runs over all boundary conditions, again as described above and $j$ runs over all indices of twig blobs.
+The index $\overline{c}$ runs over all boundary conditions,
-The final $\lf(X \setmin B^t; c^t)$ must be interpreted as fields which are splittable along all of the blobs in $\overline{B}$.
+again as described above and $j$ 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
 \[
 	\bd : \bc_k(X) \to \bc_{k-1}(X)
 \]
 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 $B_{j+1},\ldots,B_{k}$.
 If $B_j$ is a twig blob, we have to assign new local relation labels
-if removing $B_j$ creates new twig blobs. \todo{Have to say what happens when no new twig blobs are created}
+if removing $B_j$ creates new twig blobs.
+\todo{Have to say what happens when no new twig blobs are created}
+\nn{KW: I'm confused --- why isn't it OK as written?}
 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=1}^{k} (-1)^{j+1} E_j(b).
 }
 The $(-1)^{j+1}$ factors imply that the terms of $\bd^2(b)$ all cancel.
 Thus we have a chain complex.
-Note that Property \ref{property:functoriality}, that the blob complex is functorial with respect to homeomorphisms, is immediately obvious from the definition.
+Note that Property \ref{property:functoriality}, that the blob complex is functorial with respect to homeomorphisms,
+is immediately obvious from the definition.
 A homeomorphism acts in an obvious way on blobs and on fields.
-At this point, it is time to pay back our debt and define certain notions more carefully.
+\nn{end relocated informal def}
-In order to precisely state the general definition, we'll need a suitable notion of cutting up a manifold into balls.
-\begin{defn}
-\label{defn:gluing-decomposition}
-A \emph{gluing decomposition} of an $n$-manifold $X$ is a sequence of manifolds $M_0 \to M_1 \to \cdots \to M_m = X$ such that $M_0$ is a disjoint union of balls, and each $M_k$ is obtained from $M_{k-1}$ by gluing together some disjoint pair of homeomorphic $n{-}1$-manifolds in the boundary of $M_{k-1}$.
-\end{defn}
-Given a gluing decomposition $M_0 \to M_1 \to \cdots \to M_m = X$, we say that a field is splittable along it if it is the image of a field on $M_0$.
-By `a ball in $X$' we don't literally mean a submanifold homeomorphic to a ball, but rather the image of a map from the pair $(B^n, S^{n-1})$ into $X$, which is an embedding on the interior. The boundary of a ball in $X$ is the image of a locally embedded $n{-}1$-sphere. \todo{examples, e.g. balls which actually look like an annulus, but we remember the boundary}
-\begin{defn}
-\label{defn:ball-decomposition}
-A \emph{ball decomposition} of an $n$-manifold $X$ is a collection of balls in $X$, such that there exists some gluing decomposition $M_0  \to \cdots \to M_m = X$ so that the balls are the images of the components of $M_0$ in $X$.
-\end{defn}
-In particular, the union of all the balls in a ball decomposition comprises all of $X$. \todo{example}
-We'll now slightly restrict the possible configurations of blobs.
-\begin{defn}
-\label{defn:configuration}
-A configuration of $k$ blobs in $X$ is an ordered collection of $k$ balls in $X$ such that there is some gluing decomposition $M_0  \to \cdots \to M_m = X$ of $X$ and each of the balls is the image of some connected component of one of the $M_k$. Such a gluing decomposition is \emph{compatible} with the configuration.
-\end{defn}
-In particular, this implies what we said about blobs above: that for any two blobs in a configuration of blobs in $X$, they either have disjoint interiors, or one blob is strictly contained in the other. We describe these as disjoint blobs and nested blobs. Note that nested blobs may have boundaries that overlap, or indeed coincide. Blobs may meet the boundary of $X$.
-Note that the boundaries of a configuration of $k$ blobs may cut up the manifold $X$ into components which are not themselves manifolds. \todo{example: the components between the boundaries of the balls may be pathological}
-In the informal description above, in the definition of a $k$-blob diagram we asked for any collection of $k$ balls which were pairwise disjoint or nested. We now further insist that the balls are a configuration in the sense of Definition \ref{defn:configuration}. Also, we asked for a local relation on each twig blob, and a field on the complement of the twig blobs; this is unsatisfactory because that complement need not be a manifold. Thus, the official definition is
-\begin{defn}
-\label{defn:blob-diagram}
-A $k$-blob diagram on $X$ consists of
-\begin{itemize}
-\item a configuration of $k$ blobs in $X$,
-\item and a field $r \in \cC(X)$ which is splittable along some gluing decomposition compatible with that configuration,
-\end{itemize}
-such that
-the restriction of $r$ to each twig blob $B_i$ lies in the subspace $U(B_i) \subset \cC(B_i)$.
-\end{defn}
-and
-\begin{defn}
-\label{defn:blobs}
-The $k$-th vector space $\bc_k(X)$ of the \emph{blob complex} of $X$ is the direct sum of all configurations of $k$ blobs in $X$ of the vector space of $k$-blob diagrams with that configuration, modulo identifying the vector spaces for configurations that only differ by a permutation of the balls by the sign of that permutation. The differential $bc_k(X) \to bc_{k-1}(X)$ is, as above, the signed sum of ways of forgetting one ball from the configuration, preserving the field $r$.
-\end{defn}
-We readily see that if a gluing decomposition is compatible with some configuration of blobs, then it is also compatible with any configuration obtained by forgetting some blobs, ensuring that the differential in fact lands in the space of $k{-}1$-blob diagrams.
-A slight compensation to the complication of the official definition arising from attention to splitting is that the differential now just preserves the entire field $r$ without having to say anything about gluing together fields on smaller components.
-\todo{this notion of configuration of blobs is the minimal one that allows gluing and engulfing}
 We define the {\it support} of a blob diagram $b$, $\supp(b) \sub X$,

changeset 474	6a3bc1c10586
parent 473	270164061847
child 479	cfad13b6b1e5