On null cobordism classes of quasitoric manifolds and their small covers

摘要

Let Omega(U)(s) (resp. 91) 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 tau is (un-oriented) cobordant to zero in 912n if and only if its small cover M-tau is (un-oriented) cobordant to zero in n(n). 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 tau. 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 n(12), 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 : Omega(U)(12) - n(12). (c) 2020 Elsevier B.V. All rights reserved.