Spaces of polytopes and cobordism of quasitoric manifolds

Buchstaber, V. and Panov, T. and Ray, N. (2007) Spaces of polytopes and cobordism of quasitoric manifolds. Moscow Mathematical Journal, 7 (2). pp. 219-242. ISSN 1609-4514

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Abstract

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.

Item Type:	Article
Uncontrolled Keywords:	Analogous polytopes, complex cobordism, connected sum, framing, omniorientation, quasitoric manifold, stable tangent bundle
Subjects:	MSC 2010, the AMS's Mathematics Subject Classification > 14 Algebraic geometry MSC 2010, the AMS's Mathematics Subject Classification > 52 Convex and discrete geometry MSC 2010, the AMS's Mathematics Subject Classification > 55 Algebraic topology
Depositing User:	Ms Lucy van Russelt
Date Deposited:	21 Nov 2007
Last Modified:	20 Oct 2017 14:12
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/939

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