nLab
Stiefel-Whitney class

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

A family of characteristic classes that obstruct orientation, spin structure, spin^c structure, orientation of EO(2)-theory etc.

Stiefel-Whitney classes have coefficients in 2\mathbb{Z}_2, but via the Bockstein homomorphism they are lifted to integral Stiefel-Whitney classes.

Definition

Definition

(axiomatic definition)

The Stiefel-Whitney classes are characteristic classes w iH i(BO(n), 2)w_i \in H^{i}(B O(n), \mathbb{Z}_2) on the classifying space of the orthogonal group in dimension nn, defined by

  1. w 0=1w_0 = 1 and if i>ni \gt n then w i=0w_i = 0;

  2. for n=1n = 1, w 10w_1 \neq 0;

  3. for the inclusion ι:BO(n)BO(n1)\iota : B O(n) \hookrightarrow B O(n-1) we have ι *w i (n+1)=w i (n)\iota^* w_i^{(n+1)} = w_i^{(n)};

  4. sum rule: for all k,lk,l \in \mathbb{N} with the canonical inclusion

    ι:BO(k)×BO(l)BO(k+l) \iota : B O(k) \times B O(l) \to B O(k+l)

    we have for all ii \in \mathbb{N} that

    ι *w i= j=0 iw jw ij \iota^* w_i = \sum_{j = 0}^i w_j \cup w_{i-j}

    (on the right the cup product).

Definition

For EXE \to X a real vector bundle/orthogonal group-principal bundle, the total universal Stiefel-Whitney class w(E)w(E) is

w(E) iw i(E)H (X, 2) w(E) \coloneqq \sum_i w_i(E) \in H^\bullet(X, \mathbb{Z}_2)

as an element in the cohomology ring.

Remark

For the total SW class of def. 2, the sum rule of def. 1 says equivalently that for E 1,E 2E_1, E_2 two real vector bundles, then the total SW class of their direct sum of vector bundles is the cup product of the separate classes:

w(E 1E 2)=w(E 1)w(E 2). w(E_1 \oplus E_2) = w(E_1) \cup w(E_2) \,.

Constructions

(…)

Properties

Spanning the cohomology ring

Proposition

Every class in H (BO(n), 2)H^\bullet(B O(n), \mathbb{Z}_2) can be written uniquely as a polynomial in the Stiefel-Whitney classes. In fact the cohomology ring is the polynomial algebra over 2\mathbb{Z}_2 in the SW classes:

H (BO(n), 2) 2[w 1,,w n]. H^\bullet(B O(n), \mathbb{Z}_2) \simeq \mathbb{Z}_2[w_1, \cdots, w_n] \,.

Whitney duality formula

For X qX \hookrightarrow \mathbb{R}^q an embedding of a compact manifold, write τ:=TX\tau := T X for the tangent bundle and ν\nu for the corresponding normal bundle. Then since

τνT q \tau \oplus \nu \simeq T \mathbb{R}^q

and the class of the vector bundle on the right is trivial, the sum rule for the SW classes says gives the cup product duality

w(τ)w(ν)=1. w(\tau) \cup w(\nu) = 1 \,.

Relation to Chern-classes

Proposition

If E E_{\mathbb{C}} is a complex vector bundle/ U(n)U(n)-principal bundle and E E_{\mathbb{R}} is the underlying real vector bundle / O(2n)O(2n)-principal bundle then the second Stiefel-Whitney class is given by the first Chern class mod 2:

w 2(E )=c 1(E )mod2. w_2(E_{\mathbb{R}}) = c_1(E_{\mathbb{C}}) \; mod 2 \,.

This is discussed at Spin^c-striucture – From almost complex structures.

Remark

More generally, the SW classes are then given by the Chern character. See for instance Milnor-Stasheff, p. 171.

References

A classic work on the subject is

A concise introduction is in chapter 23, section 3 of

See also

Revised on October 28, 2014 09:14:05 by Urs Schreiber (141.0.9.62)