Projective product spaces

TL;DR

This paper studies the topological and geometric properties of projective product spaces, including their cohomology, immersion dimension, and parallelizability, extending understanding of these complex manifolds.

Contribution

It provides explicit calculations of cohomology rings, Steenrod algebra actions, and conditions for parallelizability and immersion dimension of projective product spaces.

Findings

Computed integral cohomology rings of P_nbar

Determined when P_nbar is parallelizable

Established formulas for immersion dimension depending on n_i

Abstract

Let nbar=(n_1,...,n_r). The quotient space P_nbar:=(S^{n_1} x...x S^{n_r})/(x ~ -x)is what we call a projective product space. We determine the integral cohomology ring and the action of the Steenrod algebra. We give a splitting of Sigma P_nbar in terms of stunted real projective spaces, and determine when S^{n_i} is a product factor. We relate the immersion dimension and span of P_nbar to the much-studied sectioning question for multiples of the Hopf bundle over real projective spaces. We show that the immersion dimension of P_nbar depends only on min(n_i), sum n_i, and r, and determine its precise value unless all n_i exceed 9. We also determine exactly when P_nbar is parallelizable.