equal
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inserted
replaced
154 on $X$ gives rise to a permissible configuration on $X\sgl$. |
154 on $X$ gives rise to a permissible configuration on $X\sgl$. |
155 (This is necessary for Proposition \ref{blob-gluing}.) |
155 (This is necessary for Proposition \ref{blob-gluing}.) |
156 \end{itemize} |
156 \end{itemize} |
157 Combining these two operations can give rise to configurations of blobs whose complement in $X$ is not |
157 Combining these two operations can give rise to configurations of blobs whose complement in $X$ is not |
158 a manifold. |
158 a manifold. |
159 Thus will need to be more careful when speaking of a field $r$ on the complement of the blobs. |
159 Thus we will need to be more careful when speaking of a field $r$ on the complement of the blobs. |
160 |
160 |
161 \begin{example} \label{sin1x-example} |
161 \begin{example} \label{sin1x-example} |
162 Consider the four subsets of $\Real^3$, |
162 Consider the four subsets of $\Real^3$, |
163 \begin{align*} |
163 \begin{align*} |
164 A & = [0,1] \times [0,1] \times [0,1] \\ |
164 A & = [0,1] \times [0,1] \times [0,1] \\ |
206 %is the image a ball, with embedded interior and possibly glued-up boundary; |
206 %is the image a ball, with embedded interior and possibly glued-up boundary; |
207 %distinct blobs should either have disjoint interiors or be nested; |
207 %distinct blobs should either have disjoint interiors or be nested; |
208 %and the entire configuration should be compatible with some gluing decomposition of $X$. |
208 %and the entire configuration should be compatible with some gluing decomposition of $X$. |
209 \begin{defn} |
209 \begin{defn} |
210 \label{defn:configuration} |
210 \label{defn:configuration} |
211 A configuration of $k$ blobs in $X$ is an ordered collection of $k$ subsets $\{B_1, \ldots B_k\}$ |
211 A configuration of $k$ blobs in $X$ is an ordered collection of $k$ subsets $\{B_1, \ldots, B_k\}$ |
212 of $X$ such that there exists a gluing decomposition $M_0 \to \cdots \to M_m = X$ of $X$ and |
212 of $X$ such that there exists a gluing decomposition $M_0 \to \cdots \to M_m = X$ of $X$ and |
213 for each subset $B_i$ there is some $0 \leq l \leq m$ and some connected component $M_l'$ of |
213 for each subset $B_i$ there is some $0 \leq l \leq m$ and some connected component $M_l'$ of |
214 $M_l$ which is a ball, so $B_i$ is the image of $M_l'$ in $X$. |
214 $M_l$ which is a ball, so $B_i$ is the image of $M_l'$ in $X$. |
215 We say that such a gluing decomposition |
215 We say that such a gluing decomposition |
216 is \emph{compatible} with the configuration. |
216 is \emph{compatible} with the configuration. |
236 this is unsatisfactory because that complement need not be a manifold. Thus, the official definitions are |
236 this is unsatisfactory because that complement need not be a manifold. Thus, the official definitions are |
237 \begin{defn} |
237 \begin{defn} |
238 \label{defn:blob-diagram} |
238 \label{defn:blob-diagram} |
239 A $k$-blob diagram on $X$ consists of |
239 A $k$-blob diagram on $X$ consists of |
240 \begin{itemize} |
240 \begin{itemize} |
241 \item a configuration $\{B_1, \ldots B_k\}$ of $k$ blobs in $X$, |
241 \item a configuration $\{B_1, \ldots, B_k\}$ of $k$ blobs in $X$, |
242 \item and a field $r \in \cF(X)$ which is splittable along some gluing decomposition compatible with that configuration, |
242 \item and a field $r \in \cF(X)$ which is splittable along some gluing decomposition compatible with that configuration, |
243 \end{itemize} |
243 \end{itemize} |
244 such that |
244 such that |
245 the restriction $u_i$ of $r$ to each twig blob $B_i$ lies in the subspace |
245 the restriction $u_i$ of $r$ to each twig blob $B_i$ lies in the subspace |
246 $U(B_i) \subset \cF(B_i)$. |
246 $U(B_i) \subset \cF(B_i)$. |