cohomology

# Contents

## Definition

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

induces a fiber sequence

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

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

## Examples

### Third integral SW class

The third integral Stiefel-Whitney class ${W}_{3}\left(TX\right)$ of the tangent bundle of an oriented $n$-dimensional manifold $X$ vanishes if and only if the second Stiefel-Whitney class ${w}_{2}\left(TX\right)$ is in the image of the reduction mod 2 morphism

${H}^{2}\left(X;ℤ\right)\to {H}^{2}\left(X;ℤ/2ℤ\right)\phantom{\rule{thinmathspace}{0ex}}.$H^2(X;\mathbb{Z})\to H^2(X;\mathbb{Z}/2\mathbb{Z}) \,.

Since ${H}^{2}\left(X;ℤ\right)$ classifies isomorphism classes of $U\left(1\right)$-principal bundles over $X$ and ${W}_{3}\left(TX\right)$ 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 $U\left(1\right)$-bundle on $X$ “killing” the second Stiefel-Whitney class of $X$.

In particular, when ${w}_{2}\left(TX\right)$ is killed by the trivial $U\left(1\right)$-bundle, i.e., when ${w}_{2}\left(TX\right)=0$, then $X$ has a spin structure.

Revised on October 18, 2011 21:39:11 by Urs Schreiber (131.174.22.10)