  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 $t^{\mathrm{th}}$ 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.
