On null cobordism classes of quasitoric manifolds and their small covers

Abstract

Let Ω⁎U (resp. N⁎) denote the ring of unitary cobordism classes of all unitary manifolds (resp. the ring of un-oriented cobordism classes of smooth closed manifolds). In this paper, we first show that a quasitoric manifold M of dimension 2n equipped with a natural conjugation involution τ is (un-oriented) cobordant to zero in N2n if and only if its small cover Mτ is (un-oriented) cobordant to zero in Nn. This will be established by investigating an important relationship between such a quasitoric manifold of dimension 2n and its small cover of dimension n that is the fixed point set of τ. As a consequence, we also show that every quasitoric manifold of dimension 12 whose associated small cover is orientable is (un-oriented) cobordant to zero in N12, and thus any omni-oriented quasitoric manifold M of dimension 12 whose associated small cover is orientable represents an element of the kernel of the natural homomorphism r:Ω12U→N12.