Stiefel-Whitney Classes and Numbers| Stiefel-Whitney Classes and Numbers |
In General
$ Def: Characteristic classes
for bundles with structure group O(n), (ξ, π:
E → B) (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 : B → B' is covered by a map ξ
→ ξ', 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(E ⊕ F) = ∑i=1k wi(E) ∪ wk − i(F) , or w(E ⊕ F) = 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 = n − k + 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(ξ
⊕ η) = 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.
* ξ ⊕ η 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.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 14 jan 2016