Projective Product Spaces

Let n = (n1, . . . , nr). The quotient space Pn := Sn1× · · ·×Snr/(x ∼ −x) is what we call a projective product space. We determine the integral cohomology ring H∗(Pn) and the action of the Steenrod algebra on H∗(Pn;Z2). We give a splitting of ΣPn in terms of stunted real projective spaces, and determine when Si is a product factor of Pn. We relate the immersion dimension and span of Pn to the much-studied sectioning question for multiples of the Hopf bundle over real projective spaces. We show that the immersion dimension of Pn depends only on min(ni), ∑ ni, and r, and determine its precise value unless all ni ≥ 10. We also determine exactly when Pn is parallelizable.