nLab integral Stiefel-Whitney class

Definition

The short exact sequence of abelian groups

$$0\to \mathbb{Z}\stackrel{\cdot2}{\to} \mathbb{Z}\to\mathbb{Z}/2\mathbb{Z}\to 0$$

induces a fiber sequence

$$\cdots\to\mathbf{B}^n \mathbb{Z}\to \mathbf{B}^n\mathbb{Z}\to \mathbf{B}^n\mathbb{Z}/2\mathbb{Z}\to \mathbf{B}^{n+1}\mathbb{Z}\to \cdots$$

and so, for any object $X$, a fiber sequence

$$\cdots\to\mathbf{H}(X,\mathbf{B}^n \mathbb{Z})\to \mathbf{H}(X,\mathbf{B}^n\mathbb{Z})\to \mathbf{H}(X,\mathbf{B}^n\mathbb{Z}/2\mathbb{Z})\stackrel{\beta_2}{\to} \mathbf{H}(X,\mathbf{B}^{n+1}\mathbb{Z})\to \cdots$$

of cocycle ∞-groupoid (with respect to any ambient (∞,1)-topos $\mathbf{H}$, such as Top $\simeq$ ∞Grpd), where $\beta_2$ is the Bockstein morphism asociated with the multiplication by 2.

The image via $\beta_2$ of the $n$-th Stiefel-Whitney map $w_n\in \mathbf{H}(X,\mathbf{B}^n\mathbb{Z}/2\mathbb{Z})$ in $\mathbf{H}(X,\mathbf{B}^{n+1}\mathbb{Z})$ is called the $(n+1)$st integral Stiefel-Whithey map and is denoted by $W_{n+1}$.

One usually uses the same symbol to denote the image of this characteristic map in cohomology (on connected components ) of $W_{n+1}$ in $H^{n+1}(X;\mathbb{Z})=\pi_0\mathbf{H}(X,\mathbf{B}^{n+1}\mathbb{Z})$, and calls this the $(n+1)$-th integral Stiefel-Whitney class.

Examples

Third integral SW class

The third integral Stiefel-Whitney class $W_3(T X)$ of the tangent bundle of an oriented $n$-dimensional manifold $X$ vanishes if and only if the second Stiefel-Whitney class $w_2(T X)$ is in the image of the reduction mod 2 morphism

$$H^2(X;\mathbb{Z})\to H^2(X;\mathbb{Z}/2\mathbb{Z}) \,.$$

Since $H^2(X;\mathbb{Z})$ classifies isomorphism classes of $U(1)$-principal bundles over $X$ and $W_3(T X)$ is the obstruction to the existence of a spin^c structure on $X$, we see that $X$ has a $spin^c$ structure if and only if there exists a principal $U(1)$-bundle on $X$ “killing” the second Stiefel-Whitney class of $X$.

In particular, when $w_2(T X)$ is killed by the trivial $U(1)$-bundle, i.e., when $w_2(T X)=0$, then $X$ has a spin structure.

The vanishing of the third integral SW class, hence spin^c-structure is the orientation condition in complex K-theory $KU$ 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.

Related concepts

• integral Steenrod square

• integral Wu structure

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 $KU$	third integral SW class $W_3$	spinᶜ structure	K-theoretic geometric quantization	Freed-Witten anomaly
2	EO(n)	Stiefel-Whitney class $w_4$
2	integral Morava K-theory $\tilde K(2)$	seventh integral SW class $W_7$			Diaconescu-Moore-Witten anomaly in Kriz-Sati interpretation

References

• Gerd Rudolph, Matthias Schmidt, around Def. 4.2.20 of Differential Geometry and Mathematical Physics: Part II. Fibre Bundles, Topology and Gauge Fields, Theoretical and Mathematical Physics series, Springer 2017 (doi:10.1007/978-94-024-0959-8)