Special and general types
The short exact sequence of abelian groups
induces a fiber sequence
and so, for any object , a fiber sequence
of cocycle ∞-groupoid (with respect to any ambient (∞,1)-topos , such as Top ∞Grpd), where is the Bockstein morphism asociated with the multiplication by 2.
The image via of the -th Stiefel-Whitney map in is called the st integral Stiefel-Whithey map and is denoted by .
One usually uses the same symbol to denote the image of this characteristic map in cohomology (on connected components ) of in , and calls this the -th integral Stiefel-Whitney class.
Third integral SW class
The third integral Stiefel-Whitney class of the tangent bundle of an oriented -dimensional manifold vanishes if and only if the second Stiefel-Whitney class is in the image of the reduction mod 2 morphism
Since classifies isomorphism classes of -principal bundles over and is the obstruction to the existence of a spin^c structure on , we see that has a structure if and only if there exists a principal -bundle on “killing” the second Stiefel-Whitney class of .
In particular, when is killed by the trivial -bundle, i.e., when , then has a spin structure.
The vanishing of the third integral SW class, hence spin^c-structure is the orientation condition in complex K-theory over oriented manifolds. In the context of string theory this is also known as the Freed-Witten anomaly cancellation condition.
Seventh integral SW class
Analogously, the vanishing of the seventh integral SW class is essentially the condition for orientation in second integral Morava K-theory.
In the context of string theory this is also known as the Diaconescu-Moore-Witten anomaly cancellation condition.
| 1 | complex K-theory | third integral SW class | spin^c-structure | K-theoretic geometric quantization | Freed-Witten anomaly | | 2 | EO(n) | Stiefel-Whitney class | | | | | 2 | integral Morava K-theory | seventh integral SW class | | | Diaconescu-Moore-Witten anomaly in Kriz-Sati interpretation |