Stiefel-Whitney Classes and Numbers  

In General
$ Def: Characteristic classes for bundles with structure group O(n), (ξ, π: EB) (fiber dimension n), with \(\mathbb Z_2\) coefficients, defined by the axioms
(1) The class wi(ξ) ∈ Hi(B; \(\mathbb Z_2\)), i ∈ \(\mathbb N\), with w0(ξ) = 1 and wi(ξ) = 0, for all i > n.
(2) (Naturality) If f : BB' is covered by a map ξξ', then wi(ξ) = f *wi(ξ').
(3) (Whitney product theorem) If E and F are two bundles over the same base B, then the Stiefel-Whitney classes, or, more compactly, the total class are

wk(EF) = ∑i=1k wi(E) ∪ wki(F) ,   or   w(EF) = w(E) ∪ w(F) .

(4) (They are not all trivial) The class \(w_1^~(\gamma_1^{~1}\)) ≠ 0, where \(\gamma_1^{~1}\) is the open Möbius strip.
$ Total Stiefel-Whitney class: For a bundle ξ, it is w(ξ):= 1 + w1(ξ) + w2(ξ) + ... ∈ H*(B; \(\mathbb Z_2\)).
@ References: in Milnor & Stasheff 74.
> Online resources: see Wikipedia page.

Special Cases
* When the base space dimension is even, they coincide with the Euler class.
* \(\mathbb R\)n-bundle: In general, all n classes may be non-zero; However, if ξ is a \(\mathbb R^n\)-bundle with k nowhere-dependent cross-sections, then wi(ξ) = 0 for i = nk + 1, ..., n.
* For a tangent bundle TM, the w(TM)s are topological invariants of M.
* For a trivial bundle ξ, wi(ξ) = 0, for all i > 0, and wi(ξη) = wi(η), for all i and η.
* w(TM) = 0 iff M is orientable.
* w(TM) ≠ 0 iff M has no spin structure.

Other Properties
* ξ ~ η iff wi(ξ) = wi(η), for all i.
* ξη is trivial iff w(η) can be expressed in terms of w(ξ), as w(η) = w(ξ)−1 [> see in particular the Whitney Duality theorem].

Examples\(\def\CP{{\mathbb C}{\rm P}}\)
* w(T(\(\mathbb R\)Pn)) = (1+a)n+1, where a is a generator of H1(T(\(\mathbb R\)Pn); \(\mathbb Z\)2).
* w1(T(\(\CP^n\))) = 0, w2(T(\(\CP^n\))) = 0 for n odd, and = x for n even, where x is a generator of H2(\(\mathbb C\)Pn; \(\mathbb Z\)2).
* w(TSn) = 1 (i.e., same as for the trivial bundle).
* w(\(\gamma_n^{~1}\)) = 1 + a, where a is a generator of H1(\(\mathbb R\)Pn; \(\mathbb Z\)2).

Applications > s.a. characteristic classes.
* And physics: The first two are used to establish whether a manifold admits a spin structure, and one can define spinor fields on it (this has been known since the 1960s); The third one is related to chirality; The (vanishing of the) highest Stiefel-Whitney class of a spacetime manifold is related to stable causality.
@ 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 = \(w_1^~\)(TM)r1 \(w_2^~\)(TM)r2 ··· \(w_n^~\)(TM)rn ∈ Hn(M; \(\mathbb Z\)2) ,

with ri ≥ 0 and ∑i i ri = n, is defined by

w1r1 w2r2 ··· \(w_n^~\)rn [M]:= \(\langle\)c, μM\(\rangle\) ,

where μM is the fundamental homology class of M.
* And bordism: Two closed n-manifolds M and N are bordant if and only if all their Stiefel-Whitney numbers agree [@ Thom CMH(54)].
* And boundaries: All Stiefel-Whitney numbers of a manifold M vanish iff M is the boundary of some smooth compact manifold.

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