168 \end{align*} |
168 \end{align*} |
169 Here $A \cup B = [0,1] \times [-1,1] \times [0,1]$ and $C \cup D = [-1,0] \times [-1,1] \times [0,1]$. |
169 Here $A \cup B = [0,1] \times [-1,1] \times [0,1]$ and $C \cup D = [-1,0] \times [-1,1] \times [0,1]$. |
170 Now, $\{A\}$ is a valid configuration of blobs in $A \cup B$, |
170 Now, $\{A\}$ is a valid configuration of blobs in $A \cup B$, |
171 and $\{D\}$ is a valid configuration of blobs in $C \cup D$, |
171 and $\{D\}$ is a valid configuration of blobs in $C \cup D$, |
172 so we must allow $\{A, D\}$ as a configuration of blobs in $[-1,1]^2 \times [0,1]$. |
172 so we must allow $\{A, D\}$ as a configuration of blobs in $[-1,1]^2 \times [0,1]$. |
173 Note however that the complement is not a manifold. |
173 Note however that the complement is not a manifold. See Figure \ref{fig:blocks}. |
174 \end{example} |
174 \end{example} |
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175 |
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176 \begin{figure}[t]\begin{equation*} |
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177 \mathfig{.4}{definition/blocks} |
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178 \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} |
175 |
179 |
176 \begin{defn} |
180 \begin{defn} |
177 \label{defn:gluing-decomposition} |
181 \label{defn:gluing-decomposition} |
178 A \emph{gluing decomposition} of an $n$-manifold $X$ is a sequence of manifolds |
182 A \emph{gluing decomposition} of an $n$-manifold $X$ is a sequence of manifolds |
179 $M_0 \to M_1 \to \cdots \to M_m = X$ such that each $M_k$ is obtained from $M_{k-1}$ |
183 $M_0 \to M_1 \to \cdots \to M_m = X$ such that each $M_k$ is obtained from $M_{k-1}$ |