K-theory of real Grassmann manifolds

TL;DR

This paper computes the complex K-theory ring of real Grassmann manifolds using spectral sequences, providing a comprehensive description for many cases and complete results under specific modular conditions.

Contribution

It applies Hodgkin spectral sequence techniques to determine the complex K-ring of real Grassmann manifolds, extending previous knowledge with new calculations and partial classifications.

Findings

Computed the complex K-ring of G_{n,k} for all relevant n,k

Achieved complete results when n ≡ 0 mod 4 and k ≡ 1 mod 2

Provided partial calculations with small indeterminacy

Abstract

Let $G_{n,k}$ denote the real Grassmann manifold of $k$-dimensional vector subspaces of $\mathbb R^n$. Using the Hodgkin spectral sequence, we compute the complex $K$-ring of $G_{n,k}$, up to a small indeterminacy, for all values of $n,k$ where $2\le k\le n-2$. When $n\equiv 0\!\!\mod 4, k\equiv 1\!\!\mod 2$ our result is complete.