Bott periodicity for inclusions of symmetric spaces

TL;DR

This paper explores the extension of Bott periodicity from classical groups to inclusions of symmetric spaces, revealing geometric periodicity phenomena and their implications for homotopy theory.

Contribution

It demonstrates that Bott periodicity extends to standard inclusions of symmetric spaces, broadening the understanding of geometric periodicity phenomena.

Findings

Bott periodicity extends to inclusions of symmetric spaces.

Geometric periodicity phenomena are observed in these inclusions.

Applications to homotopy theory are discussed.

Abstract

When looking at Bott's original proof of his periodicity theorem for the stable homotopy groups of the orthogonal and unitary groups, one sees in the background a differential geometric periodicity phenomenon. We show that this geometric phenomenon extends to the standard inclusion of the orthogonal group into the unitary group. Standard inclusions between other classical Riemannian symmetric spaces are considered as well. An application to homotopy theory is also discussed.