Fixed point sets of deformations of polyhedra with local cut points

Abstract:

A locally finite simplicial complex $X$ is said to be 2-dimensionally connected if $X - \{\text {local cut points of } X \}$ is connected. Such spaces exhibit “classical” behavior in that they all admit deformations with one fixed point, and they admit fixed point free deformations if and only if the Euler characteristic is zero. A result of G.-H. Shi implies that, for non 2-dimensionally connected spaces, the fixed point sets of deformations are equivalent to the fixed point sets of certain combinatorial maps which he calls good displacements. U. K. Scholz combined Shi’s results with a theorem of P. Hall to obtain a characterization of all finite simplicial complexes which admit fixed point free deformations. In this paper we begin by explicitly capturing the combinatorial structure of a non 2-dimensionally connected polyhedron in a bipartite graph. We then apply an extended version of Hall’s theorem to this graph to get a realization theorem which gives necessary and sufficient conditions for the existence of a deformation with a prescribed finite fixed point set. Scholz’s result, and a characterization of all finite simplicial complexes without fixed point free deformations that admit deformations with a single fixed point follow immediately from this realization theorem.