Cohomology and K-theory of generalized Dold manifolds fibred by complex flag manifolds

TL;DR

This paper computes the cohomology and K-theory of generalized Dold manifolds fibred by complex flag manifolds, revealing their algebraic structures and extending results to broader classes of spaces with involutions.

Contribution

It determines the additive and partial ring structures of cohomology and K-theory for these manifolds, generalizing previous results to complex flag bundles and wider involution spaces.

Findings

Computed the additive cohomology structure of P(m,ν) over Z

Determined the ring structure of cohomology with invertible 2

Almost completely determined the additive K-theory of P(m,ν)

Abstract

Let $\nu=(n_1,\ldots, n_s), s\ge 2,$ be a sequence of positive integers and let $n=\sum_{1\le j\le s}n_j$. Let $\mathbb CG(\nu)=U(n)/(U(n_1)\times \cdots\times U(n_s))$ be the complex flag manifold. Denote by $P(m,\nu)=P(\mathbb S^m,\mathbb CG(\nu))$ the generalized Dold manifold $\mathbb S^m\times \mathbb CG(\nu)/\langle \theta\rangle $ where $\theta=\alpha\times \sigma$ with $\alpha:\mathbb S^m\to \mathbb S^m$ being the antipodal map and $\sigma:\mathbb CG(\nu)\to \mathbb CG(\nu)$, the complex conjugation. The manifold $P(m,\nu)$ has the structure of a smooth $\mathbb CG(\nu)$-bundle over the real projective space $\mathbb RP^m.$ We determine the additive structure of $H^*(P(m,\nu);R)$ when $R=\mathbb Z$ and its ring structure when $R$ is a commutative ring in which $2$ is invertible. As an application, we determine the additive structure of $K(P(m,\nu))$ almost completely and also obtain partial results on its ring structure. The results for the singular homology are obtained for generalized Dold spaces $P(S,X)=S\times X/\langle \theta\rangle$, where $\theta=\alpha\times \sigma$, $\alpha:S\to S$ is a fixed point free involution and $\sigma:X\to X$ is an involution with $\mathrm{Fix}(\sigma)\ne \emptyset,$ for a much wider class of spaces $S$ and $X$.