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

2024/07/04 by Manas K. Mandal, Mandal, Manas, Parameswaran Sankaran +1 · 1 citation
Mathematics · #Advanced Topics in Algebra #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2407.03932

Abstract

Let ν=(n₁,…, n_s), s≥ 2, be a sequence of positive integers and let n=∑_{1≤ j≤ s}n_j. Let \mathbb CG(ν)=U(n)/(U(n₁)× ⋯× U(n_s)) be the complex flag manifold. Denote by P(m,ν)=P(\mathbb S^m,\mathbb CG(ν)) the generalized Dold manifold \mathbb S^m× \mathbb CG(ν)/⟨ θ⟩ where θ=α× σ with α:\mathbb S^m→ \mathbb S^m being the antipodal map and σ:\mathbb CG(ν)→ \mathbb CG(ν), the complex conjugation. The manifold P(m,ν) has the structure of a smooth \mathbb CG(ν)-bundle over the real projective space \mathbb RP^m. We determine the additive structure of H^*(P(m,ν);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,ν)) 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× X/⟨ θ⟩, where θ=α× σ, α:S→ S is a fixed point free involution and σ:X→ X is an involution with Fix(σ)≠ ∅, for a much wider class of spaces S and X.

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