Homological critical points for hypergraphs and simplicial complexes

Abstract

In this paper, we study the homological critical points for persistence hypergraphs and persistence simplicial complexes. With the help of the relative homology groups, we define the homological critical points for persistence morphisms between persistence hypergraphs and persistence simplicial maps between persistence simplicial complexes. We prove some commutative diagrams of subset relations for the homological critical points of persistence hypergraphs as well as persistence morphisms between them. We also prove some commutative diagrams of subset relations for the homological critical points of persistence simplicial complexes as well as persistence simplicial maps between them. As examples, we use the parametric configuration spaces to construct the parametric independence complexes as the space of parametric packings and construct the parametric dominating hypergraphs as the space of parametric coverings.

arXiv comment33 pages