Spaces of polytopes and cobordism of quasitoric manifolds

By V. Buchstaber, T. Panov, Nigel Ray. 2006

101 citations recorded by the source corpus.

Our aim is to bring the theory of analogous polytopes to bear on the study of quasitoric manifolds, in the context of stably complex manifolds with compatible torus action. By way of application, we give an explicit construction of a quasitoric representative for every complex cobordism class as the quotient of a free torus action on a real quadratic complete intersection. We suggest a systematic description for omnioriented quasitoric manifolds in terms of combinatorial data, and explain the relationship with non-singular projective toric varieties (otherwise known as toric manifolds). By expressing the first and third authors' approach to the representability of cobordism classes in these terms, we simplify and correct two of their original proofs concerning quotient polytopes; the first relates to framed embeddings in the positive cone, and the second involves modifying the operation of connected sum to take account of orientations. Analogous polytopes provide an informative setting for several of the details.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b85de7b1cf7534c1c784dfe802ea149567c40b4a

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In Discrete Mathematics and Combinatorics

1. M. Masuda
2. T. Panov
3. D. Suh
4. Suyoung Choi
5. Yukiko Fukukawa
6. Hiroaki Ishida
7. V. Buchstaber
8. Nigel Ray

In Geometry and Topology

1. M. Masuda
2. D. Suh
3. Suyoung Choi
4. T. Panov
5. Yukiko Fukukawa
6. Hiroaki Ishida
7. V. Buchstaber
8. Nigel Ray

Citations in the corpus

• Topological toric manifolds (2010, background): “5], the paper [3] treats quasitoric manifolds where the acting group is a compact torus; so we shall give the proof below for convenience sake of the readers.”
• Classification problems of toric manifolds via topology (2007, background): “a characteristic function on P. It is known that any characteristic function on P can be realized as the characteristic function of some quasitoric manifold over P (see [ 5, Section 3 ] 1 ).”
• Semifree circle actions, Bott towers and quasitoric manifolds (2006, background): “(A canonical equivariant smooth structure coinciding with the standard one in the toric case is described in [11], § 4.”
• Quasitoric manifolds over a product of simplices (2008, background): “1, there exists a unique (up to equivalence) small cover (when d D 1) and quasitoric manifold (when d D 2) with as the characteristic function, see [5] or [2] for details.”