induces a fiber sequence
and so, for any object , a fiber sequence
The image via of the -th Stiefel-Whitney map in is called the st integral Stiefel-Whithey map and is denoted by .
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.
|chromatic level||generalized cohomology theory / E-∞ ring||obstruction to orientation in generalized cohomology||generalized orientation/polarization||quantization||incarnation as quantum anomaly in higher gauge theory|
|1||complex K-theory||third integral SW class||spin^c-structure||K-theoretic geometric quantization||Freed-Witten anomaly|
|2||integral Morava K-theory||seventh integral SW class||Diaconescu-Moore-Witten anomaly in Kriz-Sati interpretation|