Simplicial Homeology and Homeotopy

TL;DR

This paper introduces new algebraic invariants called homeology and homeotopy groups for finite simplicial complexes, which are invariant under homeomorphisms and provide a finer classification than homotopy types.

Contribution

It defines the concepts of homeology, cohomeology, and homeotopy groups, establishing their invariance under homeomorphisms and introducing the homeotopy type as a new classification tool.

Findings

Homeology and cohomeology groups are invariants of polyhedra.

Homeotopy type is a new classification finer than homotopy but coarser than homeomorphism.

The paper proves invariance of these groups under homeomorphisms.

Abstract

In this paper, we define homeology group, reduced homeology group, cohomeology group and reduced cohomeology group on finite simpicial complexes and prove that these groups are homeomorphism invariants of polyhedra. We also define homeotopy type of polyhedra which is finer than homotopy type but coarser than homeomorphism class.