A study of topology of the flip Stiefel manifolds

A well known quotient of the real Stiefel manifold is the projective Stiefel manifold. We introduce a new family of quotients of the real Stiefel manifold by cyclic group of order 2 whose action is induced by simultaneous pairwise flipping of the coordinates. We obtain a description for their tangent bundles, compute their mod 2 cohomology and compute the Stiefel Whitney classes of these manifolds. We use these to provide applications to their stable span, parallelizability and equivariant maps, and some associated results in topological combinatorics.