On the cohomology of torus manifolds
Mikiya Masuda, Taras Panov

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…
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Geometric and Algebraic Topology · Algebraic Geometry and Number Theory
