On the cohomology of torus manifolds

By M. Masuda, T. Panov. 2003

171 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/3e156204b1a1b23c5c8ddd4ad176301081b0f80d

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

1. T. Panov
2. M. Masuda
3. V. Buchstaber
4. Suyoung Choi
5. D. Suh
6. Yukiko Fukukawa
7. Hiroaki Ishida
8. Hanchul Park

In Discrete Mathematics and Combinatorics

1. M. Masuda
2. T. Panov
3. V. Buchstaber
4. D. Suh
5. Suyoung Choi
6. Nigel Ray
7. Yukiko Fukukawa
8. Hiroaki Ishida

In Geometry and Topology

1. M. Masuda
2. T. Panov
3. D. Suh
4. V. Buchstaber
5. Suyoung Choi
6. Nigel Ray
7. Yukiko Fukukawa
8. Hiroaki Ishida

Citations in the corpus

• Topological toric manifolds (2010, methodology): “The following theorem from [14] clarifies the relation between the topology of a torus manifold M and the topology and combinatorics of the orbit space M/(S(1)).”
• Classification problems of toric manifolds via topology (2007, background): “See [8], [9] and [25] for further study on Bott towers.”
• Semifree circle actions, Bott towers and quasitoric manifolds (2006, unknown): “is surjective ([18], Lemma 2.”
• Torus graphs and simplicial posets (2005, background): “The question of identifying the class of simplicial posets which arise as P(Γ) for some torus graph Γ might be a difficult one, although our next statement sheds some light on this problem.”
• On the cohomology and their torsion of real toric objects (2013, background): “It should be noted that the formulas for the integral cohomology ring of toric objects have been well established in many literatures such as [8], [9], [12].”
• Toric Topology (2012, background): “Homological aspects of this study which we present here were developed in [209] and [204].”
• Categorical aspects of toric topology (2007, background): “Such constructions involve the categories cat(K) in a local rôle, and are already well-publicised.”
• On Quasitoric Orbifolds (2008, unknown): “We point out that there are even dimensional compact manifolds with locally standard T n action for which the orbit space is not a homology polytope; see Example 4.3 of [12].”