On generalized Dold manifolds

TL;DR

This paper introduces generalized Dold manifolds constructed from manifolds with involutions, providing formulas for their characteristic classes, criteria for stable parallelizability, and cobordism properties, with applications to complex flag manifolds.

Contribution

It develops a formula for the Stiefel-Whitney polynomial of generalized Dold manifolds and establishes criteria for their stable parallelizability and cobordism class, extending previous understanding.

Findings

Derived a formula for the Stiefel-Whitney polynomial of P(m,X).

Provided criteria for stable parallelizability of P(m,X).

Established conditions for the (non)vanishing of cobordism classes.

Abstract

Let $X$ be a smooth manifold with a (smooth) involution $\sigma:X\to X$ such that $Fix(\sigma)\ne \emptyset$. 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.