I know that if the link of a simplex $\sigma$ in a finite simplicial complex $K$ is contractible then the two complexes $K$ and $K\setminus \text{Star}(\sigma)$ share the same homotopy type.
Basically, if the link of $\sigma$ is contractible then it can be reduced to a point by a sequence of collapses and anti-collapses and, from this sequence, one can deduces a sequence of collapses and anti-collapses that reduces $K$ to $K\setminus \text{Star}(\sigma)$.
I am wondering if there is some similar result with homology. Namely, if the link of $\sigma$ in $K$ is acyclic, is it true that $K$ and $K\setminus \text{Star}(\sigma)$ have the same homology groups?
I would really appreciate any idea, reference or counter-example!