On generalized Dold manifolds

Abstract

Let $X$ be a smooth manifold with a (smooth) involution $\sigma :X\to X$ such that $Fix(\sigma )\ne \varnothing$. We call the space $P(m,X):={\mathbb{S}}^m\times X/\sim$ where $(v,x)\sim (-v,\sigma (x))$ a generalized Dold manifold. When $X$ is an almost complex manifold and the differential $T\sigma :TX\to TX$ 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.

arXiv comment19 pages. A minor error in Prop. 2.5(iii) had been corrected. There was a gap in the proof of Theorem 1.2 which has been corrected. Other minor typos were corrected

References in corpus (1)

• Nonexistence of Almost Complex Structures on the product $S^{2m}\times M$