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

Abstract

Let $\nu =(n_1,\dots ,n_s),s\ge 2,$ be a sequence of positive integers and let $n={\sum }_{1\le j\le s}{n}_j$. Let $\mathbb{C}G(\nu )=U(n)/(U(n_1)\times \cdots \times U(n_s))$ be the complex flag manifold. Denote by $P(m,\nu )=P({\mathbb{S}}^m,\mathbb{C}G(\nu ))$ the generalized Dold manifold ${\mathbb{S}}^m\times \mathbb{C}G(\nu )/⟨\theta ⟩$ where $\theta =\alpha \times \sigma$ with $\alpha :{\mathbb{S}}^m\to {\mathbb{S}}^m$ being the antipodal map and $\sigma :\mathbb{C}G(\nu )\to \mathbb{C}G(\nu )$, the complex conjugation. The manifold $P(m,\nu )$ has the structure of a smooth $\mathbb{C}G(\nu )$-bundle over the real projective space $\mathbb{R}{P}^m.$ We determine the additive structure of $H^{\ast }(P(m,\nu );R)$ 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 $P(S,X)=S\times X/⟨\theta ⟩$, where $\theta =\alpha \times \sigma$, $\alpha :S\to S$ is a fixed point free involution and $\sigma :X\to X$ is an involution with $\mathrm{Fix}(\sigma )\ne \varnothing ,$ for a much wider class of spaces $S$ and $X$.