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}%

   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}