Stiefel-Whitney Classes and Numbers

In General
$ Def: Characteristic classes for bundles with structure group O(n), (, : E → B) (fiber dim n), with Z₂ coefficients, defined by the axioms
(1) The class w_i() Hⁱ(B; Z₂), i N, with w₀() = 1 and w_i() = 0, for all i > n.
(2) (Naturality) If f : B → B' is covered by a map → ', then w_i() = f *w_i(').
(3) (Whitney product theorem) If E and F are two bundles over the same base space B, then the Stiefel-Whitney classes, or, more compactly, the total class are

w_k(E F) = _i=1^k w_i(E) w_{k – i}(F) ,   or   w(E F) = w(E) w(F) .

(4) (They are not all trivial) The class w₁(₁¹) 0, where ₁¹ is the open Möbius strip.
$ Total Stiefel-Whitney class: For a bundle , it is w():= 1 + w₁() + w₂() + ... H*(B; Z₂).
* Applications: They are used to establish whether a manifold admits a spin structure.
@ References: in Milnor & Stasheff 74.

Special Cases
* When the base space dimension is even, they coincide with the Euler class.
* Rⁿ-bundle: In general, all n classes may be nonzero; However, if is a Rⁿ-bundle with k nowhere-dependent cross-sections, then w_i() = 0 for i = n – k + 1, ..., n.
* For a tangent bundle TM, the w(TM)'s are topological invariants of M.
* For a trivial bundle , w_i() = 0, for all i > 0, and w_i( ) = w_i(), for all i and .
* w(TM) = 0 iff M is orientable.
* w(TM) 0 iff M has no spin structure.

Other Properties
* iff w_i() = w_i(), for all i.
* is trivial iff w() can be expressed in terms of w(), as w() = w()^–1 [> see in particular the Whitney Duality theorem].

Examples
* w(T(RPⁿ)) = (1+a)ⁿ⁺¹, where a is a generator of H¹(T(RPⁿ); Z₂).
* w₁(T(CPⁿ)) = 0, w₂(T(CPⁿ)) = 0 for n odd, and = x for n even, where x is a generator of H²(CPⁿ; Z₂).
* w(TSⁿ) = 1 (i.e., same as for the trivial bundle).
* w(_n¹) = 1 + a, where a is a generator of H¹(RPⁿ; Z₂).

Other Topics and References > see characteristic classes.
* And physics: The first two are related to the existence of spinor fields (this has been known since the 1960s), the third one to chirality.
@ And physics: Nielsen Flagga & Antonsen IJTP(02) [spin and chirality], IJTP(04) [causality].

Stiefel-Whitney Numbers
$ Def: Given a manifold M of dimension n, the Stiefel-Whitney number associated to any monomial

c = w₁(TM)^r_1 w₂(TM)^r_2 ··· w_n(TM)^r_n Hⁿ(M; Z₂) ,

with r_i 0 and _i i r_i = n, is defined by

w₁^r_1 w₂^r_2 ··· w_n^r_n [M]:= c, _M ,

where _M is the fundamental homology class of M.
* Special cases:
- If n is odd, all Stiefel-Whitney numbers vanish;
- If n is even, at least w_n[M] and w₁ⁿ[M] are non-vanishing.
- All Stiefel-Whitney numbers vanish iff M is the boundary of some smooth compact manifold.

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