Bott Periodicity, Submanifolds, and Vector Bundles

TL;DR

This paper provides a geometric proof of the Atiyah-Bott-Shapiro theorem linking Clifford modules to vector bundles over spheres, avoiding the use of Bott periodicity by employing explicit deformations.

Contribution

It offers a direct geometric proof of the theorem, establishing the correspondence without relying on Bott periodicity, using explicit deformation techniques.

Findings

Clifford modules correspond to vector bundles over spheres

The proof is constructed via explicit deformations

The approach bypasses Bott periodicity theorem

Abstract

We sketch a geometric proof of the classical theorem of Atiyah, Bott, and Shapiro \cite{ABS} which relates Clifford modules to vector bundles over spheres. Every module of the Clifford algebra $Cl_k$ defines a particular vector bundle over $\S^{k+1}$, a generalized Hopf bundle, and the theorem asserts that this correspondence between $Cl_k$-modules and stable vector bundles over $\S^{k+1}$ is an isomorphism modulo $Cl_{k+1}$-modules. We prove this theorem directly, based on explicit deformations as in Milnor's book on Morse theory \cite{M}, and without referring to the Bott periodicity theorem as in \cite{ABS}.