nLab Atiyah-Bott-Shapiro isomorphism

Idea

The Atiyah-Bott-Shapiro isomorphism (due to Atiyah-Bott-Shapiro 63) is an isomorphism between the real/complex topological K-theory ring of the point in degree $q$ and the quotient of Clifford modules of rank $q$ by those that have an extension to Clifford modules of rank $q+1$.

The generalization of this to Clifford module bundles is the content of Karoubi K-theory.

Related concepts

• Karoubi K-theory

• K-orientation

References

The orginal reference is

• Michael Atiyah, Raoul Bott, Arnold Shapiro, Clifford modules, Topology 3(Suppl 1):3–38 (1963) (pdf)

The result is reviewed as theorem I 9.27 in

• H. Blaine Lawson, Marie-Louise Michelsohn, chapter I, section 9 of Spin geometry, Princeton University Press (1989)