Mikiya Masuda

On the cohomology of torus manifolds

A torus manifold is an even-dimensional manifold acted on by a half-dimensional torus with non-empty fixed point set and some additional orie ntation 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 geo metry: 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.

Quasitoric manifolds over a product of simplices

A quasitoric manifold (resp. a small cover) is a

Rigidity Problems in Toric Topology: A Survey

2n

Cohomological non-rigidity of generalized real Bott manifolds of height 2

-dimensional (resp. an

CLASSIFICATION OF REAL BOTT MANIFOLDS AND ACYCLIC DIGRAPHS

n

Stably trivial equivariant algebraic vector bundles

-dimensional) smooth closed manifold with an effective locally standard action of

Buchstaber invariants of skeleta of a simplex

(S^1)^n

ELLIPTIC GENERA, TORUS ORBIFOLDS AND MULTI-FANS II

(resp.

Tangential representations of cyclic group actions on homotopy complex projective spaces

(\mathbb Z_2)^n

The equivariant cohomology rings of Peterson varieties

) whose orbit space is combinatorially an