On the homotopy types of $4$-dimensional toric orbifolds

TL;DR

This paper investigates the relationship between cohomology and homotopy types of 4-dimensional toric orbifolds, introducing proper isomorphisms and showing they classify at most two homotopy types per class.

Contribution

It reformulates the cohomological rigidity problem for 4D toric orbifolds using proper isomorphisms and establishes bounds on the number of homotopy types within each class.

Findings

Each proper isomorphism class contains at most two homotopy types.

The two classifications coincide in certain special cases.

Abstract

The cohomological rigidity problem for toric orbifolds asks when an integral cohomology isomorphism implies a homotopy equivalence. In this paper we reformulate the cohomological rigidity problem in the context of $4$-dimensional toric orbifolds by introducing what we call proper isomorphisms, a variant of a concept studied by J.H.C. Whitehead. We prove that each proper isomorphism class of $4$-dimensional toric orbifolds contains at most two distinct homotopy types, and that the two classifications agree in certain special circumstances.