Simplicial (co)homeology groups: New P.L. homeomorphism invariants of polyhedra

TL;DR

This paper introduces new simplicial (co)homeology groups as PL homeomorphism invariants that capture more geometric information than traditional (co)homology, and defines the finer homeotopy type of polyhedra.

Contribution

It defines (reduced) (co)homeology groups for simplicial complexes and proves they are PL homeomorphism invariants, providing new tools for geometric classification.

Findings

(Co)homeology groups are PL homeomorphism invariants.

These groups are not homotopy invariants.

They are invariants of the finer homeotopy type.

Abstract

In this paper, we define (reduced) homeology groups and (reduced) cohomeology groups on finite simpicial complexes and prove that these groups are PL homeomorphsm invariants of polyhedra, while they are not homotopy invariants. So these groups can reflect some information that (co)homology groups can not tell. We also define homeotopy type of polyhedra which is finer than homotopy type but coarser than homeomorphism class, and prove that (co)homeology groups are actually homeotopy invariants. In the last section of this paper, we give a geometric description of some special (co)homeology groups.