Bott Periodicity, Submanifolds, and Vector Bundles
Jost Eschenburg, Bernhard Hanke

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 defines a particular vector bundle over , a generalized Hopf bundle, and the theorem asserts that this correspondence between -modules and stable vector bundles over is an isomorphism modulo -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}.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Geometric and Algebraic Topology · Advanced Operator Algebra Research
