Osaka Journal of Mathematics

Quasitoric manifolds over a product of simplices

Suyoung Choi, Mikiya Masuda, and Dong Youp Suh

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Abstract

A quasitoric manifold (resp. a small cover) is a $2n$-dimensional (resp. an $n$-dimensional) smooth closed manifold with an effective locally standard action of $(S^{1})^{n}$ (resp. $(\mathbb{Z}_{2})^{n}$) whose orbit space is combinatorially an $n$-dimensional simple convex polytope $P$. In this paper we study them when $P$ is a product of simplices. A generalized Bott tower over $\mathbb{F}$, where $\mathbb{F}=\mathbb{C}$ or $\mathbb{R}$, is a sequence of projective bundles of the Whitney sum of $\mathbb{F}$-line bundles starting with a point. Each stage of the tower over $\mathbb{F}$, which we call a generalized Bott manifold, provides an example of quasitoric manifolds (when $\mathbb{F}=\mathbb{C}$) and small covers (when $\mathbb{F}=\mathbb{R}$) over a product of simplices. It turns out that every small cover over a product of simplices is equivalent (in the sense of Davis and Januszkiewicz [5]) to a generalized Bott manifold. But this is not the case for quasitoric manifolds and we show that a quasitoric manifold over a product of simplices is equivalent to a generalized Bott manifold if and only if it admits an almost complex structure left invariant under the action. Finally, we show that a quasitoric manifold $M$ over a product of simplices is homeomorphic to a generalized Bott manifold if $M$ has the same cohomology ring as a product of complex projective spaces with $\mathbb{Q}$ coefficients.

Article information

Source
Osaka J. Math., Volume 47, Number 1 (2010), 109-129.

Dates
First available in Project Euclid: 19 February 2010

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1266586788

Mathematical Reviews number (MathSciNet)
MR2666127

Zentralblatt MATH identifier
1237.57036

Subjects
Primary: 57S15: Compact Lie groups of differentiable transformations 14M25: Toric varieties, Newton polyhedra [See also 52B20]
Secondary: 57S25: Groups acting on specific manifolds

Citation

Choi, Suyoung; Masuda, Mikiya; Suh, Dong Youp. Quasitoric manifolds over a product of simplices. Osaka J. Math. 47 (2010), no. 1, 109--129. https://projecteuclid.org/euclid.ojm/1266586788


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References

  • V.M. Buchstaber and T.E. Panov: Torus Actions and Their Applications in Topology and Combinatorics, University Lecture Series 24, Amer. Math. Soc., Providence, RI, 2002.
  • V.M. Buchstaber, T.E. Panov and N. Ray: Spaces of polytopes and cobordism of quasitoric manifolds, Mosc. Math. J. 7 (2007), 219--242, arXiv:math/0609346v2.
  • S. Choi, M. Masuda and D.Y. Suh: Topological classification of generalized Bott towers, Trans. Amer. Math. Soc. 362 (2010), 1097--1112, arXiv:0807.4334.
  • Y. Civan and N. Ray: Homotopy decompositions and $K$-theory of Bott towers, $K$-Theory 34 (2005), 1--33.
  • M.W. Davis and T. Januszkiewicz: Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J. 62 (1991), 417--451.
  • N.È. Dobrinskaya: Classification problem for quasitoric manifolds over a given simple polytope, Funct. Anal. and Appl. 35 (2001), 83--89.
  • M. Grossberg and Y. Karshon: Bott towers, complete integrability, and the extended character of representations, Duke Math. J. 76 (1994), 23--58.
  • A. Hattori and T. Yoshida: Lifting compact group actions in fiber bundles, Japan. J. Math. (N.S.) 2 (1976), 13--25.
  • M. Masuda: Unitary toric manifolds, multi-fans and equivariant index, Tohoku Math. J. (2) 51 (1999), 237--265.
  • M. Masuda and T.E. Panov: Semifree circle actions, Bott towers, and quasitoric manifolds, Sbornik Math. 199 (2008), 1201--1223, arXiv:math.AT/0607094.