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2007.35: Tangential structures on toric manifolds, and connected sums of polytopes

2007.35: Victor M. Buchstaber and Nigel Ray (2001) Tangential structures on toric manifolds, and connected sums of polytopes. International Mathematics Research Notices, 2001. pp. 193-219. ISSN 1073-7928

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DOI: 10.1155/S1073792801000125

Abstract

We extend work of Davis and Januszkiewicz by considering omnioriented toric manifolds, whose canonical codimension-2 submanifolds are independently oriented. We show that each omniorientation induces a canonical stably complex structure, which is respected by the torus action and so defines an element of an equivariant cobordism ring. As an application, we compute the complex bordism groups and cobordism ring of an arbitrary omnioriented toric manifold. We consider a family of examples Bi,j, which are toric manifolds over products of simplices, and verify that their natural stably complex structure is induced by an omniorientation. Studying connected sums of products of the Bi,j allows us to deduce that every complex cobordism class of dimension >2 contains a toric manifold, necessarily connected, and so provides a positive answer to the toric analogue of Hirzebruch's famous question for algebraic varieties. In previous work, we dealt only with disjoint unions, and ignored the relationship between the stably complex structure and the action of the torus. In passing, we introduce a notion of connected sum # for simple n-dimensional polytopes; when Pn is a product of simplices, we describe Pn#Qn by applying an appropriate sequence of pruning operators, or hyperplane cuts, to Qn.

Item Type:Article
Subjects:MSC 2000 > 55 Algebraic topology
MSC 2000 > 57 Manifolds and cell complexes
MIMS number:2007.35
Deposited By:Miss Louise Stait
Deposited On:29 March 2007

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