## Osaka Journal of Mathematics

### On generalized Dold manifolds

#### Abstract

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

#### Article information

Source
Osaka J. Math., Volume 56, Number 1 (2019), 75-90.

Dates
First available in Project Euclid: 16 January 2019

https://projecteuclid.org/euclid.ojm/1547607627

Mathematical Reviews number (MathSciNet)
MR3908778

#### Citation

Nath, Avijit; Sankaran, Parameswaran. On generalized Dold manifolds. Osaka J. Math. 56 (2019), no. 1, 75--90. https://projecteuclid.org/euclid.ojm/1547607627

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