# The Betti numbers of a triangulated ball relative to a disk in its boundary

Fix $n \geq 1$ and let $B$ denote a triangulated closed $n$-ball. Let $D$ be a subset of the boundary of $B^n$ that is homeomorphic to the closed $(n-1)$-ball and such that is properly triangulated by the same triangulation as well.

I would like to compute the Betti numbers of $B$ relative to $D$. I have tried to regard the long exact sequence on homology of the chain complexes, and that the $0$-th homology is trivial as it is isomorphic to the $0$-th homology group of the de-Rham complex. Is it correct that all Betti number of $B$ relative to $D$ vanish?

I am not used to algebraic topology, and need the result only in a secondary (if even that) context. I could use a reference which clarifies under which circumstances to use the long exact sequence on homology in the context of simplicial complexes.

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