Exotic torus manifolds and equivariant smooth structures on quasitoric manifolds

TL;DR

This paper investigates the topological and smooth classification of torus manifolds and quasitoric manifolds, revealing distinctions between homotopy, homeomorphism, and diffeomorphism types, especially in six dimensions.

Contribution

It provides new insights into the classification of torus and quasitoric manifolds, including the existence of non-homeomorphic homotopy equivalent examples and the enumeration of conjugacy classes of tori.

Findings

Homotopy equivalent torus manifolds may not be homeomorphic.

Characterization of fundamental groups of locally standard torus manifolds.

In six dimensions, the number of conjugacy classes of tori is finite, unlike higher dimensions.

Abstract

In 2006 Masuda and Suh asked if two compact non-singular toric varieties having isomorphic cohomology rings are homeomorphic. In the first part of this paper we discuss this question for topological generalizations of toric varieties, so-called torus manifolds. For example we show that there are homotopy equivalent torus manifolds which are not homeomorphic. Moreover, we characterize those groups which appear as the fundamental groups of locally standard torus manifolds. In the second part we give a classification of quasitoric manifolds and certain six-dimensional torus manifolds up to equivariant diffeomorphism. In the third part we enumerate the number of conjugacy classes of tori in the diffeomorphism group of torus manifolds. For torus manifolds of dimension greater than six there are always infinitely many conjugacy classes. We give examples which show that this does not hold for six-dimensional torus manifolds.