On generalized Dold manifolds

Abstract

Let X be a smooth manifold with a (smooth) involution σ : X → X such that Fix(σ) = ∅. We call the space P (m, X) := S m × X/ ∼ where (v, x) ∼ (−v, σ(x)) a generalized Dold manifold. When X is an almost complex manifold and the differential T σ : T X → T X is conjugate complex linear on each fibre, we obtain a formula for the Stiefel-Whitney polynomial of P (m, X) when H 1 (X; Z 2 ) = 0. We obtain results on stable parallelizability of P (m, X) and a very general criterion for the (non) vanishing of the unoriented cobordism class [P (m, X)] in terms of the corresponding properties for X. These results are applied to the case when X is a complex flag manifold.