155 Thus will need to be more careful when speaking of a field $r$ on the complement of the blobs. |
155 Thus will need to be more careful when speaking of a field $r$ on the complement of the blobs. |
156 |
156 |
157 \begin{example} |
157 \begin{example} |
158 Consider the four subsets of $\Real^3$, |
158 Consider the four subsets of $\Real^3$, |
159 \begin{align*} |
159 \begin{align*} |
160 A & = [0,1] \times [0,1] \times [-1,1] \\ |
160 A & = [0,1] \times [0,1] \times [0,1] \\ |
161 B & = [0,1] \times [-1,0] \times [-1,1] \\ |
161 B & = [0,1] \times [-1,0] \times [0,1] \\ |
162 C & = [-1,0] \times \setc{(y,z)}{z^2 \sin(1/z) \leq y \leq 1, z \in [-1,1]} \\ |
162 C & = [-1,0] \times \setc{(y,z)}{z \sin(1/z) \leq y \leq 1, z \in [0,1]} \\ |
163 D & = [-1,0] \times \setc{(y,z)}{-1 \leq y \leq z^2 \sin(1/z), z \in [-1,1]}. |
163 D & = [-1,0] \times \setc{(y,z)}{-1 \leq y \leq z \sin(1/z), z \in [0,1]}. |
164 \end{align*} |
164 \end{align*} |
165 Here $A \cup B = [0,1] \times [-1,1] \times [-1,1]$ and $C \cup D = [-1,0] \times [-1,1] \times [-1,1]$. Now, $\{A\}$ is a valid configuration of blobs in $A \cup B$, and $\{C\}$ is a valid configuration of blobs in $C \cup D$, so we must allow $\{A, C\}$ as a configuration of blobs in $[-1,1]^3$. Note however that the complement is not a manifold. |
165 Here $A \cup B = [0,1] \times [-1,1] \times [0,1]$ and $C \cup D = [-1,0] \times [-1,1] \times [0,1]$. Now, $\{A\}$ is a valid configuration of blobs in $A \cup B$, and $\{C\}$ is a valid configuration of blobs in $C \cup D$, so we must allow $\{A, C\}$ as a configuration of blobs in $[-1,1]^2 \times [0,1]$. Note however that the complement is not a manifold. |
166 \end{example} |
166 \end{example} |
167 |
167 |
168 \begin{defn} |
168 \begin{defn} |
169 \label{defn:gluing-decomposition} |
169 \label{defn:gluing-decomposition} |
170 A \emph{gluing decomposition} of an $n$-manifold $X$ is a sequence of manifolds |
170 A \emph{gluing decomposition} of an $n$-manifold $X$ is a sequence of manifolds |
172 by gluing together some disjoint pair of homeomorphic $n{-}1$-manifolds in the boundary of $M_{k-1}$. |
172 by gluing together some disjoint pair of homeomorphic $n{-}1$-manifolds in the boundary of $M_{k-1}$. |
173 If, in addition, $M_0$ is a disjoint union of balls, we call it a \emph{ball decomposition}. |
173 If, in addition, $M_0$ is a disjoint union of balls, we call it a \emph{ball decomposition}. |
174 \end{defn} |
174 \end{defn} |
175 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 $M_0$. |
175 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 $M_0$. |
176 |
176 |
177 In the example above, note that $$A \sqcup B \sqcup C \sqcup D \to (A \cup B) \sqcup (C \cup D) \to [-1,1]^3$$ is a ball decomposition of $[-1,1]^3$, but other sequences of gluings from $A \sqcup B \sqcup C \sqcup D$ to $[-1,1]^3$ have intermediate steps which are not manifolds. |
177 In the example above, note that $$A \sqcup B \sqcup C \sqcup D \to (A \cup B) \sqcup (C \cup D) \to A \cup B \cup C \cup D$$ is a ball decomposition, but other sequences of gluings starting from $A \sqcup B \sqcup C \sqcup D$have intermediate steps which are not manifolds. |
178 |
178 |
179 We'll now slightly restrict the possible configurations of blobs. |
179 We'll now slightly restrict the possible configurations of blobs. |
180 \begin{defn} |
180 \begin{defn} |
181 \label{defn:configuration} |
181 \label{defn:configuration} |
182 A configuration of $k$ blobs in $X$ is an ordered collection of $k$ subsets $\{B_1, \ldots B_k\}$ of $X$ such that there exists a gluing decomposition $M_0 \to \cdots \to M_m = X$ of $X$ and for each subset $B_i$ there is some $0 \leq r \leq m$ and some connected component $M_r'$ of $M_r$ which is a ball, so $B_i$ is the image of $M_r'$ in $X$. Such a gluing decomposition is \emph{compatible} with the configuration. A blob $B_i$ is a twig blob if no other blob $B_j$ maps into the appropriate $M_r'$. \nn{that's a really clumsy way to say it, but I struggled to say it nicely and still allow boundaries to intersect -S} |
182 A configuration of $k$ blobs in $X$ is an ordered collection of $k$ subsets $\{B_1, \ldots B_k\}$ of $X$ such that there exists a gluing decomposition $M_0 \to \cdots \to M_m = X$ of $X$ and for each subset $B_i$ there is some $0 \leq r \leq m$ and some connected component $M_r'$ of $M_r$ which is a ball, so $B_i$ is the image of $M_r'$ in $X$. Such a gluing decomposition is \emph{compatible} with the configuration. A blob $B_i$ is a twig blob if no other blob $B_j$ maps into the appropriate $M_r'$. \nn{that's a really clumsy way to say it, but I struggled to say it nicely and still allow boundaries to intersect -S} |
189 Blobs may meet the boundary of $X$. |
189 Blobs may meet the boundary of $X$. |
190 Further, note that blobs need not actually be embedded balls in $X$, since parts of the boundary of the ball $M_r'$ may have been glued together. |
190 Further, note that blobs need not actually be embedded balls in $X$, since parts of the boundary of the ball $M_r'$ may have been glued together. |
191 |
191 |
192 Note that often the gluing decomposition for a configuration of blobs may just be the trivial one: if the boundaries of all the blobs cut $X$ into pieces which are all manifolds, we can just take $M_0$ to be these pieces, and $M_1 = X$. |
192 Note that often the gluing decomposition for a configuration of blobs may just be the trivial one: if the boundaries of all the blobs cut $X$ into pieces which are all manifolds, we can just take $M_0$ to be these pieces, and $M_1 = X$. |
193 |
193 |
194 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 |
194 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 definitions are |
195 \begin{defn} |
195 \begin{defn} |
196 \label{defn:blob-diagram} |
196 \label{defn:blob-diagram} |
197 A $k$-blob diagram on $X$ consists of |
197 A $k$-blob diagram on $X$ consists of |
198 \begin{itemize} |
198 \begin{itemize} |
199 \item a configuration $\{B_1, \ldots B_k\}$ of $k$ blobs in $X$, |
199 \item a configuration $\{B_1, \ldots B_k\}$ of $k$ blobs in $X$, |
200 \item and a field $r \in \cC(X)$ which is splittable along some gluing decomposition compatible with that configuration, |
200 \item and a field $r \in \cC(X)$ which is splittable along some gluing decomposition compatible with that configuration, |
201 \end{itemize} |
201 \end{itemize} |
202 such that |
202 such that |
203 the restriction $u_i$ of $r$ to each twig blob $B_i$ lies in the subspace $U(B_i) \subset \cC(B_i)$. |
203 the restriction $u_i$ of $r$ to each twig blob $B_i$ lies in the subspace $U(B_i) \subset \cC(B_i)$. (See Figure \ref{blobkdiagram}.) More precisely, each twig blob $B_i$ is the image of some ball $M_r'$ as above, and it is really the restriction to $M_r'$ that must lie in the subspace $U(M_r')$. |
204 \end{defn} |
204 \end{defn} |
205 \todo{Careful here: twig blobs aren't necessarily balls?} |
|
206 (See Figure \ref{blobkdiagram}.) |
|
207 \begin{figure}[t]\begin{equation*} |
205 \begin{figure}[t]\begin{equation*} |
208 \mathfig{.7}{definition/k-blobs} |
206 \mathfig{.7}{definition/k-blobs} |
209 \end{equation*}\caption{A $k$-blob diagram.}\label{blobkdiagram}\end{figure} |
207 \end{equation*}\caption{A $k$-blob diagram.}\label{blobkdiagram}\end{figure} |
210 and |
208 and |
211 \begin{defn} |
209 \begin{defn} |