nLab dual Stiefel-Whitney class

Idea

Dual Stiefel-Whitney classes are $\mathbb{Z}_2$-valued characteristic classes for real vector bundles. The total Stiefel-Whitney class is invertible in the cohomology ring (using $w_0=1$ and the geometric series) and the dual Stiefel-Whitney classes are then the components of its inverse, meaning in particular that they can be expressed by Stiefel-Whitney classes again and carry the exact same information. Nonetheless, dual Stiefel-Whitney classes can be used to simplfy calculations. An important application is the description of embedding problems.

Definition

Let $E\twoheadrightarrow B$ be a real vector bundle of rank $n$ with total Stiefel-Whitney class $w(E)\in H^*(B,\mathbb{Z}_2)$, then its total dual Stiefel-Whitney class $\overline{w}(E)\in H^*(B,\mathbb{Z}_2)$ is defined by:

$$w(E)\overline{w}(E) =1.$$

If the Stiefel-Whitney classes are known, then the equation can be solved inductively due to their linear and triangular form. Solving the equation in order $n$ yields:

$$\overline{w}_n(E) =-\sum_{k=1}^n w_k(E)\overline{w}_{n-k}(E).$$

Another direct computation uses the geometric series:

$$\overline{w}(E) =w(E)^{-1} =\left( 1+\sum_{k=1}^n w_k(E) \right)^{-1} =\sum_{l\in 0}^\infty\left( \sum_{k=1}^n w_k(E) \right)^l.$$

Examples

Low dual Stiefel-Whitney classes are:

$$\overline{w}_0 =1;$$

$$\overline{w}_1 =-w_1;$$

$$\overline{w}_2 =-w_2 +w_1^2.$$

Properties

For real vector bundles $E,F\twoheadrightarrow B$, one has $w(E\oplus F)=w(E)w(F)$, meaning that $w(F)=\overline{w}(E)w(E\oplus F)$. In particular, if $B$ is a compact Hausdorff space, then for every $E$ there exists a $F$ with $E\oplus F\cong\mathbb{R}^n$, (Hatcher 17, Prop. 1.4) which with $w(E\oplus F)=w(\mathbb{R}^n)=1$ makes the formula simplify to:

$$\overline{w}(E) =w(F).$$

Hence in this case the dual Stiefel-Whitney classes can be seen as the Stiefel-Whitney classes of a complement to a trivial bundle.

Using this fact, an imporant application of the dual Stiefel-Whitney classes are embedding problems. For a $n$-dimensional smooth manifold $M$ and a smooth embedding $M\hookrightarrow\mathbb{R}^{n+k}$, the Whitney embedding theorem assures one can chose $k\leq n-1$ while the dual Stiefel-Whitney classes give a lower bound. Using the standard scalar product on $\mathbb{R}^{n+k}$ and orthogonal subspaces, there is a normal bundle $N_i M\twoheadrightarrow M$ with $TM\oplus N_i M\cong\underline{\mathbb{R}^{n+k}}$ and therefore:

$$w(N_i M) =\overline{w}(TM).$$

(Milnor & Stasheff 74, p. 49)

Since Stiefel-Whitney classes vanish in orders larger than the rank of the vector bundle, $k$ must be equal or larger than the highest degree in which this equation relates non-vanishing classes. With the right side, a lower bound can therefore be determined by the (dual) Stiefel-Whitney classes of $M$, explicitly by:

$$k \geq\max\{l\in\mathbb{N}|\overline{w}_l(M)\neq 0\}.$$

Examples are easy to compute for simple smooth manifolds, for which their Stiefel-Whitney classes are known, for example real projective spaces. In fact, the real projective spaces $\mathbb{R}P^{2^n}$ has its lower bound $k\geq 2^n-1$ reach all the way up to the upper bound of the Whitney embedding theorem. (Milnor & Stasheff 74, Thrm 4.8)

Related concepts

• Segre class (dual Chern class)

References

• John Milnor, Jim Stasheff, Characteristic classes, Princeton Univ. Press (1974) (ISBN:9780691081229, doi:10.1515/9781400881826, pdf)

• Allen Hatcher: Vector bundles and K-Theory, book draft (2017) [webpage, pdf]