Quasitoric Totally Normally Split Representatives in the Unitary Cobordism Ring

Abstract

The present paper generalises the results of Ray [15] and Buchstaber-Ray [1], Buchstaber-Panov-Ray [3] in unitary cobordism theory. I prove that any class x ∈ Ω * U of the unitary cobordism ring contains a quasitoric totally normally and tangentially split manifold.Second, a method of producing new totally normally split toric varieties is given in Section 3. Namely, this is blow-up of a totally normally split toric variety at an invariant complex codimension 2 subvariety.These are used to construct some totally normally split toric varieties which are then shown to be multiplicative generators of Ω * U . Finally, a possible adaptation of N. Ray's construction to Theorem 1.3 is discussed in Section 7.The author is grateful to V.M. Buchstaber and T.E. Panov for numerous fruitful discussions.2 Bounded flag fibre bundlesThe idea of the bounded flag manifold ([2], [4, §7.7]) can be globalised in terms of fiber bundles. In this Section, the corresponding construction is given. For the rest of this Section, X stands for a compact stably complex smooth manifold of real dimension 2n and ξ i → X, rk ξ i = 1, i = 1, . . . , k + 1, are complex linear vector bundles over X. Also let ξ := k+1 i=1 ξ i . Everywhere below pull-backs and tensor products of vector bundles are omitted, unless otherwise specified.

Quasitoric Totally Normally Split Manifolds

“…TTS-and TNS-manifolds appeared in the works of Arthan and Bullet [1], Ochanine and Schwartz [12], Ray [16] and [18] related to a representation of a given complex cobordism class with a manifold from a prescribed family. A naturally arising problem here is to study TTS/TNS-manifolds in well-known families of manifolds, for example, quasitoric manifolds (see [8], [9]).…”

“…Let us remind some facts about quasitoric TNS-manifolds (see [18]). The complex projective space CP n is a TNS-manifold iff n < 2.…”