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258 If an $n$-manifold $X$ contains $Y \sqcup Y^\text{op}$ as a codimension $0$ submanifold of its boundary, |
258 If an $n$-manifold $X$ contains $Y \sqcup Y^\text{op}$ as a codimension $0$ submanifold of its boundary, |
259 write $X_\text{gl} = X \bigcup_{Y}\selfarrow$ for the manifold obtained by gluing together $Y$ and $Y^\text{op}$. |
259 write $X_\text{gl} = X \bigcup_{Y}\selfarrow$ for the manifold obtained by gluing together $Y$ and $Y^\text{op}$. |
260 Note that this includes the case of gluing two disjoint manifolds together. |
260 Note that this includes the case of gluing two disjoint manifolds together. |
261 \begin{property}[Gluing map] |
261 \begin{property}[Gluing map] |
262 \label{property:gluing-map}% |
262 \label{property:gluing-map}% |
263 Given a gluing $X \to X_\mathrm{gl}$, there is |
263 Given a gluing $X \to X_\mathrm{gl}$, there is an injective natural map |
264 a natural map |
|
265 \[ |
264 \[ |
266 \bc_*(X) \to \bc_*(X_\mathrm{gl}) |
265 \bc_*(X) \to \bc_*(X_\mathrm{gl}) |
267 \] |
266 \] |
268 (natural with respect to homeomorphisms, and also associative with respect to iterated gluings). |
267 (natural with respect to homeomorphisms, and also associative with respect to iterated gluings). |
269 \end{property} |
268 \end{property} |