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integral Stiefel-Whitney class

Definition

The short exact sequence of abelian groups

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

induces a fiber sequence

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

$$\cdots \to H(X,B^n\mathbb{Z})\to H(X,B^n\mathbb{Z})\to H(X,B^n\mathbb{Z}/2\mathbb{Z})\stackrel{{\beta }_2}{\to }H(X,B^{n+1}\mathbb{Z})\to \cdots$$\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 $H$, 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 $w_n\in H(X,B^n\mathbb{Z}/2\mathbb{Z})$ in $H(X,B^{n+1}\mathbb{Z})$ 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 $H^{n+1}(X;\mathbb{Z})={\pi }_{0}H(X,B^{n+1}\mathbb{Z})$, and calls this the $(n+1)$-th integral Stiefel-Whitney class.

Examples

Third integral SW class

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

$$H^2(X;\mathbb{Z})\to H^2(X;\mathbb{Z}/2\mathbb{Z}).$$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 $W_3(TX)$ is the obstruction to the existence of a spin^c structure on $X$, we see that $X$ has a ${\mathrm{spin}}^c$ 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 $w_2(TX)$ is killed by the trivial $U(1)$-bundle, i.e., when $w_2(TX)=0$, then $X$ has a spin structure.