Homotopical computations for projective Stiefel manifolds and related quotients

Abstract

My thesis deals with various homogeneous spaces associated with the real and complex Stiefel manifolds and their homotopical computations. Primarily we work with the complex projective Stiefel manifolds. We compute their Brown-Peterson cohomology using homotopy fixed point spectral sequence and then using BP-cohomology operations provide some criteria for non-existence of an equivariant map between various complex projective Stiefel manifolds under the action of the circle group. We also study the p-local homotopy type of complex projective Stiefel manifolds and various other quotients of Stiefel manifolds and show that they admit a product decomposition into a complex projective space or lens space and some bunch of odd dimensional spheres after p-localization for all but finitely many primes p. We also calculate characteristic classes for certain quotients of Stiefel manifolds and then derive results on certain numerical invariants, such as characteristic rank, skew embedding dimensions for those spaces.