Spaces of polytopes and cobordism of quasitoric manifolds. (English) Zbl 1176.55004

Buchstaber, Victor M.; Panov, Taras E.; Ray, Nigel

Mosc. Math. J. 7, No. 2, 219-242 (2007).

In this paper, quasitoric manifolds (a useful phrase the authors use in place of the toric manifolds of Davis and Januszkiewicz which has a different different meaning from the non-singular toric varieties of algebraic geometry) are studied from the point of view of complex cobordism. In particular, the present paper fills some gaps in the proof of result of an earlier paper by two of the authors. This asserts that in all dimensions greater than 2, every complex cobordism class contains a quasitoric manifold with a certain kind of stable normal structure compatible with the associated toral action. The main new ingredient required to complete the proof is the use of analogous polytopes (originally introduced by Alexandrov in the 1930’s) to help organise the combinatorial geometry underlying the quasitoric geometry.

Reviewer: Andrew Baker (Glasgow)

MSC:

55N22	Bordism and cobordism theories and formal group laws in algebraic topology
52B20	Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
14M25	Toric varieties, Newton polyhedra, Okounkov bodies

Keywords:

analogous polytope; complex cobordism; connected sum; framing; omniorientation; quasitoric manifold