Complex-projective and lens product spaces

Abstract

Let \(t\) be a positive integer. Following work of D. M. Davis, we study the topology of complex-projective product spaces, i.e. quotients of cartesian products of odd dimensional spheres by the diagonal \(S^1\)-action, and of the \(t\)-torsion lens product spaces, i.e. the corresponding quotients when the action is restricted to the \(t\)th roots of unity. For a commutative complex-oriented cohomology theory \(h^*\), we determine the \(h^*\)-cohomology ring of these spaces (in terms of the \(t\)-series for \(h^*\), in the case of \(t\)-torsion lens product spaces). When \(h^*\) is singular cohomology with mod 2 coefficients, we also determine the action of the Steenrod algebra. We show that these spaces break apart after a suspension as a wedge of desuspensions of usual stunted complex-projective (\(t\)-torsion lens) spaces. We estimate the category and topological complexity of complex-projective and lens product spaces, showing in particular that these invariants are usually much lower than predicted by the usual dimensional bounds. We extend Davis’ analysis of manifold properties such as immersion dimension, (stable) span, and (stable) parallelizability of real-projective product spaces to the complex-projective and lens product cases.