Local homology

The local homology at $x\in X$ is $H_i(X,X\setminus \{x\})$. It is preserved by homeomorphisms and, in a simplicial complex, is computed from the reduced homology of the local link.

If a neighborhood of $x$ is a cone on a compact link $L$, excision and the long exact sequence of the cone pair give

In the two-skeleton of a barycentrically subdivided tetrahedron, the vertex corresponding to the whole tetrahedron has link equal to the connected 1-skeleton of the subdivided boundary, with 14 vertices and 36 edges. Its local $H_2$ therefore has rank $36-14+1=23$. Every other point has smaller local rank, so every self-homeomorphism fixes this vertex.