I heard that when $X$ is a connected compact topological $n$-manifold in $\mathbb{R}^n$ whose boundary $\partial X$ is a connected compact (n-1)-manifold, then we can deduce the Betti numbers of $X$ from the Betti numbers of $\partial X$. Also, we can deduce the Euler characteristic of $X$ from the one of $\partial X$.
However, I do not know if some similar results exist when we work with connected compact topological sets whose boundary is a combinatorial pseudo-manifold (that means that the boundary can own "pinches").
I would be very grateful if someone can help me on this.
Thanks for your help.