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JOURNALS // Moscow Mathematical Journal // Archive

Mosc. Math. J., 2013 Volume 13, Number 1, Pages 57–98 (Mi mmj489)

This article is cited in 24 papers

Topological toric manifolds

Hiroaki Ishidaa, Yukiko Fukukawab, Mikiya Masudab

a Osaka City University Advanced Mathematical Institute, Sumiyoshi-ku, Osaka, 558-8585, Japan
b Department of Mathematics, Osaka City University, Sumiyoshi-ku, Osaka, 558-8585, Japan

Abstract: We introduce the notion of a topological toric manifold and a topological fan and show that there is a bijection between omnioriented topological toric manifolds and complete non-singular topological fans. A topological toric manifold is a topological analogue of a toric manifold and the family of topological toric manifolds is much larger than that of toric manifolds. A topological fan is a combinatorial object generalizing the notion of a simplicial fan in toric geometry.
Prior to this paper, two topological analogues of a toric manifold have been introduced. One is a quasitoric manifold and the other is a torus manifold. One major difference between the previous notions and topological toric manifolds is that the former support a smooth action of an $S^1$-torus while the latter support a smooth action of a $\mathbb{C}^*$-torus. We also discuss their relation in details.

Key words and phrases: Toric manifold, fan, multi-fan, quasitoric manifold, torus manifold.

MSC: Primary 53D20, 57S15; Secondary 14M25

Received: August 15, 2011; in revised form January 24, 2012

Language: English

DOI: 10.17323/1609-4514-2013-13-1-57-98



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