Toric Topology and Complex Cobordism

We plan to discuss how the ideas and methodology of Toric Topology can be applied to one of the classical subjects of algebraic topology: finding nice representatives in complex cobordism classes. Toric and quasitoric manifolds are the key players in the emerging field of Toric Topology, and they constitute a sufficiently wide class of stably complex manifolds to additively generate the whole complex cobordism ring. In other words, every stably complex manifold is cobordant to a manifold with a nicely behaving torus action. An informative setting for applications of toric topology to complex cobordism is provided by the combinatorial and convex-geometrical study of analogous polytopes. 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. The latter is a yet another disguise of the moment-angle manifold, another familiar object of toric topology. 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).