Osaka Journal of Mathematics

On the cohomology of torus manifolds

Mikiya Masuda and Taras Panov

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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.

Article information

Source
Osaka J. Math., Volume 43, Number 3 (2006), 711-746.

Dates
First available in Project Euclid: 25 September 2006

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1159190010

Mathematical Reviews number (MathSciNet)
MR2283418

Zentralblatt MATH identifier
1111.57019

Subjects
Primary: 57R91: Equivariant algebraic topology of manifolds

Citation

Masuda, Mikiya; Panov, Taras. On the cohomology of torus manifolds. Osaka J. Math. 43 (2006), no. 3, 711--746. https://projecteuclid.org/euclid.ojm/1159190010


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References

  • G.E. Bredon: Introduction to Compact Transformation Groups, Academic Press, New York--London, 1972.
  • W. Bruns and J. Herzog: Cohen-Macaulay Rings, revised edition, Cambridge Studies in Adv. Math. 39, Cambridge Univ. Press, Cambridge, 1998.
  • V.M. Buchstaber and T.E. Panov: Torus Actions and Their Applications in Topology and Combinatorics, University Lecture Series 24, Amer. Math. Soc., Providence, R.I., 2002.
  • V.M. Buchstaber and N. Ray: Tangential structures on toric manifolds, and connected sums of\kern1.25pt polytopes, Internat. Math. Res. Notices 4 (2001), 193--219.
  • M.W. Davis: Groups generated by reflections and aspherical manifolds not covered by Euclidean space, Ann. of Math. 117 (1983), 293--324.
  • M.W. Davis and T. Januszkiewicz: Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J. 62 (1991), 417--451.
  • W. Fulton: An Introduction to Toric Varieties, Ann. of Math. Studies 113, Princeton Univ. Press, Princeton, N.J., 1993.
  • M. Goresky, R. Kottwitz and R. MacPherson: Equivariant cohomology, Koszul duality and the localisation theorem, Invent. Math. 131 (1998), 25--83.
  • V.W. Guillemin and S. Sternberg: Supersymmetry and Equivariant de Rham Theory, Springer, Berlin--Heidelberg--New York, 1999.
  • V.W. Guillemin and C. Zara: One-skeleta, Betti numbers and equivariant cohomology, Duke Math. J. 107 (2001), 283--349.
  • A. Hattori and M. Masuda: Theory of multi-fans, Osaka J. Math. 40 (2003), 1--68.
  • W.-Y. Hsiang: Cohomology Theory of Topological Transformation Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 85, Springer, Berlin--Heidelberg--New York, 1976.
  • M. Masuda: Unitary toric manifolds, multi-fans and equivariant index, Tohoku Math. J. 51 (1999), 237--265.
  • M. Masuda: $h$-vectors of Gorenstein$*$ simplicial posets, Adv. Math. 194 (2005), 332--344.
  • J.R. Munkres: Topological results in combinatorics, Michigan Math. J. 31 (1984), 113--128.
  • E.H. Spanier: Algebraic Topology, McGraw-Hill Book Company, New York, 1966.
  • R.P. Stanley: $f$-vectors and $h$-vectors of simplicial posets, J. Pure Appl. Algebra 71 (1991), 319--331.
  • R.P. Stanley: Combinatorics and Commutative Algebra, second edition, Progress in Math. 41, Birkhäuser, Boston, 1996.
  • R.E. Stong: Notes on Cobordism Theory, Princeton Univ. Press, Princeton, N.J., 1968.