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Moscow Mathematical Journal

Volume 13, Issue 1, January–March 2013  pp. 57–98.

Topological Toric Manifolds

Authors Hiroaki Ishida (1), Yukiko Fukukawa (2), and Mikiya Masuda (2)
Author institution: (1) Osaka City University Advanced Mathematical Institute, Sumiyoshi-ku, Osaka 558-8585, Japan
(2) Department of Mathematics, Osaka City University, Sumiyoshi-ku, Osaka 558-8585, Japan


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 S1-torus while the latter support a smooth action of a ℂ*-torus. We also discuss their relation in details.

2010 Mathematics Subject Classification. Primary: 53D20, 57S15; Secondary: 14M25.

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