On the cohomology of torus manifolds

TL;DR

This paper explores the cohomological properties of torus manifolds, linking their cohomology rings to the combinatorial structure of their orbit spaces, generalizing concepts from toric geometry.

Contribution

It establishes conditions under which the cohomology ring is generated by two-dimensional classes and characterizes the vanishing of odd-degree cohomology in relation to orbit space properties.

Findings

Cohomology ring generated by 2D classes iff orbit space is a homology polytope.

Odd-degree cohomology vanishes iff orbit space is face-acyclic.

Equivariant cohomology corresponds to face rings of simplicial complexes or posets.

Abstract

A torus manifold is an even-dimensional manifold acted on by a half-dimensional torus with non-empty fixed point set and some additional orientation data. It may be considered as a far-reaching generalisation of toric manifolds from algebraic geometry. The orbit space of a torus manifold has a rich combinatorial structure, e.g., it is a manifold with corners provided that the action is locally standard. Here we investigate relationships between the cohomological properties of torus manifolds and the combinatorics of their orbit quotients. We show that the cohomology ring of a torus manifold is generated by two-dimensional classes if and only if the quotient is a homology polytope. In this case we retrieve the familiar picture from toric geometry: the equivariant cohomology is the face ring of the nerve simplicial complex and the ordinary cohomology is obtained by factoring out certain linear forms. In a more general situation, we show that the odd-degree cohomology of a torus manifold vanishes if and only if the orbit space is face-acyclic. Although the cohomology is no longer generated in degree two under these circumstances, the equivariant cohomology is still isomorphic to the face ring of an appropriate simplicial poset.