A Study of topology of the Flip Stiefel Manifolds

TL;DR

This paper introduces new quotient manifolds of the real Stiefel manifold, analyzes their topological properties such as tangent bundles and cohomology, and explores applications in stable span, parallelizability, and topological combinatorics.

Contribution

It defines a novel family of quotient manifolds of the Stiefel manifold and computes their topological invariants, extending understanding of their geometric and combinatorial properties.

Findings

Computed mod 2 cohomology and Stiefel Whitney classes

Analyzed stable span and parallelizability

Applied results to topological combinatorics

Abstract

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 Stiefel Whitney classes of these manifolds. We use these to give applications to their stable span, parallelizability and equivariant maps, and the associated results in topological combinatorics.