Commuting matrices and Atiyah's Real K-theory

TL;DR

This paper characterizes the homotopy type of commuting n-tuples in classical groups using Real K-theory, enabling explicit calculations of related homotopy groups and coefficient rings.

Contribution

It provides a new description of the equivariant homotopy type of commuting matrices in terms of Real K-theory, leading to complete calculations of associated homotopy groups.

Findings

Homotopy groups of commuting n-tuples in the stable orthogonal group computed.

Coefficient ring for commutative orthogonal K-theory determined.

Homotopy type of commuting matrices described via Real K-theory.

Abstract

We describe the $C_2$-equivariant homotopy type of the space of commuting n-tuples in the stable unitary group in terms of Real K-theory. The result is used to give a complete calculation of the homotopy groups of the space of commuting n-tuples in the stable orthogonal group, as well as of the coefficient ring for commutative orthogonal K-theory.