The complex K ring of the flip Stiefel manifolds

TL;DR

This paper computes the complex K-theory ring of flip Stiefel manifolds, a class of quotient manifolds, for the case when the parameter s is even, using advanced algebraic topology techniques.

Contribution

It provides the first explicit calculation of the complex K-ring for flip Stiefel manifolds when s is even, addressing challenges posed by non-trivial group actions.

Findings

Calculated $K^*(FV_{m,2s})$ for s even.

Used representation theory of Spin(m) and Hodgkin spectral sequence.

Resolved the case for s ≡ 0 mod 2.

Abstract

The flip Stiefel manifolds (FV_{m,2s}) are defined as the quotient of the real Stiefel manifolds (V_{m,2s}) induced by the simultaneous pairwise flipping of the co-ordinates by the cyclic group of order 2. We calculate the complex (K)-ring of the flip Stiefel manifolds, $K^\ast(FV_{m,2s})$, for $s$ even. Standard techniques involve the representation theory of $Spin(m),$ and the Hodgkin spectral sequence. However, the non-trivial element inducing the action doesn't readily yield the desired homomorphisms. Hence, by performing additional analysis, we settle the question for the case of (s \equiv 0 \pmod 2.)