Tangential structures on toric manifolds, and connected sums of polytopes (Q2712599)

In their pioneering paper of 1991 [Duke Math. J. 62, No. 2, 417-451 (1991; Zbl 0733.52006)] M. Davis and T. Januszkiewicz defined the notion of toric manifold, which can be considered as a topological approximation to non-singular toric varieties studied in algebraic geometry. NEWLINENEWLINENEWLINEThe presents authors extend the work of Davis and Januszkiewicz by considering omnioriented toric manifolds, whose canonical codimension-2 submanifolds are independently oriented. They 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, they compute the complex bordism groups and the cobordism ring of an arbitrary omnioriented toric manifold. By considering a family of examples \(B_{i,j}\), which are toric manifolds over products of simplices, the authors verify that their natural stably complex structure is induced by an omniorientation. Studying connected sums of products of the \(B_{i,j}\) allowed them 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.