# nLab Atiyah-Bott-Shapiro isomorphism

Contents

## Spin geometry

spin geometry

Dynkin labelsp. orth. groupspin grouppin groupsemi-spin group
SO(2)Spin(2)Pin(2)
B1SO(3)Spin(3)Pin(3)
D2SO(4)Spin(4)Pin(4)
B2SO(5)Spin(5)Pin(5)
D3SO(6)Spin(6)
B3SO(7)Spin(7)
D4SO(8)Spin(8)SO(8)
B4SO(9)Spin(9)
D5SO(10)Spin(10)
B5SO(11)Spin(11)
D6SO(12)Spin(12)
$\vdots$$\vdots$
D8SO(16)Spin(16)SemiSpin(16)
$\vdots$$\vdots$
D16SO(32)Spin(32)SemiSpin(32)

string geometry

cohomology

# Contents

## 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.

## References

The orginal reference is

The result is reviewed as theorem I 9.27 in

Created on November 24, 2014 at 12:33:04. See the history of this page for a list of all contributions to it.