Equivariantly homeomorphic quasitoric manifolds are diffeomorphic

TL;DR

This paper proves that quasitoric manifolds that are equivariantly homeomorphic are also diffeomorphic, establishing a strong link between their topological and smooth structures, and showing that their diffeomorphism type is largely determined by cohomology and Pontrjagin classes.

Contribution

It demonstrates that equivariant homeomorphism implies diffeomorphism for quasitoric manifolds and clarifies the extent to which their smooth structure is determined by algebraic invariants.

Findings

Equivariantly homeomorphic quasitoric manifolds are diffeomorphic.

The diffeomorphism type is determined by cohomology rings and first Pontrjagin classes.

Finiteness results for the classification of certain quasitoric manifolds.

Abstract

In this note we prove that equivariantly homeomorphic quasitoric manifolds are diffeomorphic. As a consequence we show that up to finite ambiguity the diffeomorphism type of certain quasitoric manifolds $M$ is determined by their cohomology rings and first Pontrjagin classes.