Weighted homology theory of orbifolds and Weighted Polyhedra

TL;DR

This paper develops two new homology theories for orbifolds and weighted polyhedra that incorporate local group orders, providing invariants and duality results that extend classical homology concepts to singular spaces.

Contribution

It introduces AW-homology and DW-homology theories for orbifolds using weighted simplices, capturing structural information of singularities and generalizing classical homology.

Findings

AW-homology and DW-homology are invariants of compact orbifolds.

A natural graded product exists in DW-cohomology, generalizing the cup product.

The theories satisfy a generalized Poincaré duality for weighted polyhedra based on manifolds.

Abstract

We introduce two new homology theories of orbifolds from some special type of triangulations adapted to an orbifold, called AW-homology and DW-homology. The main idea in the definitions of these two homology theories is that we use divisibly weighted simplices as the building blocks of an orbifold and encode the orders of the local groups of the orbifold in the boundary maps of their chain complexes so that these two theories can reflect some structural information of the singular set of the orbifold. We prove that AW-homology and DW-homology groups are invariants of compact orbifolds under orbifold isomorphisms and more generally under certain type of homotopy equivalences of orbifolds. Moreover, we find that there exists a natural graded commutative product in the cohomology groups corresponding to the DW-homology, which generalizes the cup product in the ordinary simplicial cohomology. In addition, we introduce a broader class of objects called weighted polyhedra and develop our AW-homology and DW-homology theory in this broader setting. When a weighted polyhedron is based on a compact orientable homology n-manifold, we prove that its AW-homology and DW-homology satisfy a generalized version of Poincar\'e duality with respect to its DW-cohomology and AW-cohomology, respectively. Our goal is to generalize the whole simplicial (co)homology theory to any triangulable topological space with a suitable weight function.