Journal of the Mathematical Society of Japan

Locally standard torus actions and $h'$-numbers of simplicial posets

Anton AYZENBERG

Full-text: Open access

Abstract

We consider the orbit type filtration on a manifold with a locally standard torus action and study the corresponding spectral sequence in homology. When all proper faces of the orbit space are acyclic and the free part of the action is trivial, this spectral sequence can be described in full. The ranks of diagonal terms of its second page are equal to $h'$-numbers of a simplicial poset dual to the orbit space. Betti numbers of a manifold with a locally standard torus action are computed: they depend on the combinatorics and topology of the orbit space but not on the characteristic function.

A toric space whose orbit space is the cone over a Buchsbaum simplicial poset is studied by the same homological method. In this case the ranks of the diagonal terms of the spectral sequence at infinity are the $h''$-numbers of the simplicial poset. This fact provides a topological evidence for the nonnegativity of $h''$-numbers of Buchsbaum simplicial posets and links toric topology to some recent developments in enumerative combinatorics.

Article information

Source
J. Math. Soc. Japan, Volume 68, Number 4 (2016), 1725-1745.

Dates
First available in Project Euclid: 24 October 2016

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1477327231

Digital Object Identifier
doi:10.2969/jmsj/06841725

Mathematical Reviews number (MathSciNet)
MR3564449

Zentralblatt MATH identifier
1361.57030

Subjects
Primary: 57N65: Algebraic topology of manifolds
Secondary: 55R20: Spectral sequences and homology of fiber spaces [See also 55Txx] 05E45: Combinatorial aspects of simplicial complexes 06A07: Combinatorics of partially ordered sets 18G40: Spectral sequences, hypercohomology [See also 55Txx]

Keywords
locally standard torus action orbit type filtration Buchsbaum simplicial poset simplicial manifold coskeleton filtration $h'$-numbers $h''$-numbers homological spectral sequence

Citation

AYZENBERG, Anton. Locally standard torus actions and $h'$-numbers of simplicial posets. J. Math. Soc. Japan 68 (2016), no. 4, 1725--1745. doi:10.2969/jmsj/06841725. https://projecteuclid.org/euclid.jmsj/1477327231


Export citation

References

  • A. Ayzenberg, Locally standard torus actions and sheaves over Buchsbaum posets, preprint arXiv:1501.04768.
  • A. A. Ayzenberg and V. M. Buchstaber, Nerve complexes and moment-angle spaces of convex polytopes, Proc. Steklov Inst. Math., 275 (2011), 15–46.
  • V. M. Buchstaber and T. E. Panov, Combinatorics of simplicial cell complexes and torus actions, Proc. Steklov Inst. Math., 247 (2004), 33–49.
  • V. M. Buchstaber and T. E. Panov, Toric Topology, Math. Surveys Monogr., 204, Amer. Math. Soc., Providence, RI, 2015.
  • A. Cannas da Silva, V. Guillemin and A. R. Pires, Symplectic origami, IMRN 2011 (2011), 4252–4293.
  • M. W. Davis and T. Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J., 62 (1991), 417–451.
  • M. Gualtieri, S. Li, A. Pelayo and T. Ratiu, The tropical momentum map: a classification of toric log symplectic manifolds, preprint arXiv:1407.3300.
  • H. Maeda, M. Masuda and T. Panov, Torus graphs and simplicial posets, Adv. Math., 212 (2007), 458–483.
  • M. Masuda and T. Panov, On the cohomology of torus manifolds, Osaka J. Math., 43 (2006), 711–746.
  • I. Novik, Upper bound theorems for homology manifolds, Israel J. Math., 108 (1998), 45–82.
  • I. Novik and Ed Swartz, Socles of Buchsbaum modules, complexes and posets, Adv. Math., 222 (2009), 2059–2084.
  • R. P. Stanley, Combinatorics and Commutative Algebra, Second edition Progress in Math., 41, Birkhäuser, Boston, Inc., Boston, MA, 1996.
  • T. Yoshida, Local torus actions modeled on the standard representation, Adv. Math., 227 (2011), 1914–1955.