Let $X$ be a compact manifold with boundary $\partial X$ with $
\dim X\setminus \partial X=n$. Moreover, $X$ and $\partial X$ are both aspherical. Then what's the $H^n(X\cup\_{\Sigma\subset \partial X} C(\Sigma))$, where each $\Sigma $ is a boundary component? By Lefschetz duality, $H^n(X,\partial X)=\mathbb Z$. I am trying to apply the (relative) Mayer–Vietoris sequence to $H^n(X,\partial X)$ and $H^n(X\cup\_{\Sigma\subset \partial X} C(\Sigma))$ to deduce the $H^n(X\cup\_{\Sigma\subset \partial X} C(\Sigma))=\mathbb Z$, but it seems it's not very successful so far. Is there an easy way/reference to show $H^n(X\cup\_{\Sigma\subset \partial X} C(\Sigma))=\mathbb Z$?



Cohomology of the coned off space
You're going to need some more assumptions about your manifold if you want $H^n(X,\partial X)=\mathbb{Z}$ (think about connectedness and orientability).


In general (even without those assumptions), $H^n(X\cup\_{\Sigma\subset \partial X} C(\Sigma))\cong H^n(X,\partial X)$ for $n>1$:


Using the long exact sequence of the pair, $H^n(X\cup\_{\Sigma\subset \partial X} C(\Sigma))\cong H^n(X\cup\_{\Sigma\subset \partial X} C(\Sigma), \cup v\_i)$, where $v\_i$ are the cone points. Then by homotopy equivalence, this is isomorphic to $H^n(X\cup\_{\Sigma\subset \partial X} C(\Sigma), \cup C(\Sigma))$. If they weren't already, let $C(\Sigma)$ be the closed cones, and now apply excision.


I leave what happens in the cases $n=0,1$ as an exercise.


