Commuting matrices and Atiyah's 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.

arXiv commentMinor changes. To appear in Journal of Topology

References in corpus (13)

• FI-modules and stability for representations of symmetric groups
• FI-modules over Noetherian rings
• Global homotopy theory
• Vacuum structure in supersymmetric Yang-Mills theories with any gauge group
• Continuous functors as a model for the equivariant stable homotopy category
• A classifying space for commutativity in Lie groups
• Stable splittings, spaces of representations and almost commuting elements in Lie groups
• A survey on spaces of homomorphisms to Lie groups
• Infinite loop spaces and nilpotent K-theory
• On spaces of commuting elements in Lie groups
• Homological stability for spaces of commuting elements in Lie groups
• The Bott cofiber sequence in deformation K-theory and simultaneous similarity in U(n)
• The spectrum for commutative complex $K$-theory

Cited by in corpus (2)

• On the second homotopy group of spaces of commuting elements in Lie groups
• Commutative d-Torsion K-Theory and Its Applications