On generalized Dold manifolds

Let $X$ be a smooth manifold with a (smooth) involution $\sigma :X\to X$ such that $Fix(\sigma )\mathrm{\ne }\mathrm{\varnothing }$. We call the space $P(m,X):=$\mathbb{S}$^m\times X/!\sim$where$(v,x)\sim (-v,\sigma(x))$ageneralizedDoldmanifold\mathrm{.}When$X$isanalmostcomplexmanifoldandthedifferential$T\sigma: TX\to TX$ is conjugate complex linear on each fibre, we obtain a formula for the Stiefel-Whitney polynomial of $P(m,X)$ when $H^1(X;{\mathbb{Z}}_2)=0$. 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.