Fixed barycentre of the barycentric tetrahedral two-skeleton

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.