nLab Stiefel-Whitney number

Idea

Stiefel-Whitney numbers are binary topological invariants for real vector bundles over smooth manifolds, which are computed from their Stiefel-Whitney classes. In particular, they exist for smooth manifolds when considering their tangent bundle. According to the Pontrjagin-Thom theorem, two smooth manifolds are bordant if and only if all their Stiefel-Whitney numbers are identical. Two orientable smooth 5-manifolds are furthermore bordant if and only if their respective de Rham invariant, a particular Stiefel-Whitney number, is identical.

Definition

Let $E\twoheadrightarrow M$ be a real vector bundle over a $n$-dimensional manifold $M$ with a unique fundamental class $[M]\in H_n(M,\mathbb{Z}_2)\cong\mathbb{Z}_2$ and $i_1+\ldots+i_r=n$ be a partition. Using the Kronecker pairing and the cup product, the Stiefel-Whitney number of $E$ corresponding to this partition is given by:

$$w_{i_1}\ldots w_{i_r}[E] \coloneqq\langle w_{i_1}(E)\smile\ldots\smile w_{i_r}(E),[M]\rangle \in\mathbb{Z}_2.$$

Stiefel-Whitney numbers are usually considered for its tangent bundle with the short notation $w_{i_1}\ldots w_{i_r}[M]\coloneqq w_{i_1}\ldots w_{i_r}[TM]$.

(Milnor & Stasheff 74, p. 51)

Properties

Proposition

All Stiefel-Whitney numbers of a 3-manifold vanish.

(Milnor & Stasheff 74, Problem 11-D)

Theorem

(Pontrjagin theorem) All Stiefel-Whitney numbers of the boundary of a compact smooth manifold vanish.

(Milnor & Stasheff 74, Thrm. 4.9)

Theorem

(Thom theorem) If all Stiefel-Whitney numbers of a smooth manifold vanish, it is the boundary of a compact smooth manifold.

(Milnor & Stasheff 74, Thrm. 4.10)

Stiefel-Whitney numbers of a 4-manifold

A 4-manifold $M$ has five Stiefel-Whitney numbers, which are $w_1^4[M]$, $w_1^2w_2[M]$, $w_1w_3[M]$, $w_2^2[M]$ and $w_4[M]$. If $M$ is orientable, if and only if $w_1(M)=0$, then the former three vanish automatically.

Stiefel-Whitney numbers of a 5-manifold

A 5-manifold $M$ has seven Stiefel-Whitney numbers, which are $w_1^5[M]$, $w_1^3w_2[M]$, $w_1^2w_3[M]$, $w_1w_2^2[M]$, $w_1w_4[M]$, $w_2w_3[M]$ and $w_5[M]$. According to the upper properties, the last always vanishes. If $M$ is orientable, if and only if $w_1(M)=0$, then the former five vanish additionally. In this case, only the Stiefel-Whitney number $w_2w_3[M]$ can potentially be non-trivial and due to this special role, it is called de Rham-invariant.

Examples

The Stiefel-Whitney numbers of the real projective space $\mathbb{R}P^n$ are given by:

$$w_{i_1}\ldots w_{i_r}[\mathbb{R}P^n] =\binom{n+1}{i_1}\ldots\binom{n+1}{i_r}\mod 2.$$

Related concepts

• Chern number
• Pontrjagin number

References

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