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 *w*_{i}(*ξ*)
∈ H^{i}(*B*; \(\mathbb Z_2\)),
*i* ∈ \(\mathbb N\), with *w*_{0}(*ξ*)
= 1 and *w*_{i}(*ξ*)
= 0, for all *i *> *n*.

(2) (Naturality) If *f* : *B* → *B*' is covered by a map *ξ*
→ *ξ*', then *w*_{i}(*ξ*)
= *f* **w*_{i}(*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

*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 \(w_1^~(\gamma_1^{~1}\))
≠ 0, where \(\gamma_1^{~1}\) is the open Möbius strip.

$ __Total Stiefel-Whitney class__:
For a bundle *ξ*, it is *w*(*ξ*):=
1 + *w*_{1}(*ξ*)
+ *w*_{2}(*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* w*_{i}(*ξ*)
= 0 for *i* = *n* − *k* + 1, ..., *n*.

* For a tangent bundle T*M*,
the *w*(T*M*)s are topological invariants of *M*.

* For a trivial bundle *ξ*,
*w*_{i}(*ξ*) = 0, for
all *i* > 0, and *w*_{i}(*ξ*
⊕ *η*) = *w*_{i}(*η*),
for all *i* and *η*.

* *w*(T*M*) = 0
iff *M* is orientable.

* *w*(T*M*) ≠ 0
iff *M* has no spin structure.

**Other Properties**

* *ξ* ~ *η*
iff *w*_{i}(*ξ*)
= *w*_{i}(*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(\(\mathbb R\)P^{n}))
= (1+*a*)^{n+1}, where *a* is
a generator of H^{1}(T(\(\mathbb R\)P^{n});
\(\mathbb Z\)_{2}).

* *w*_{1}(T(\(\CP^n\))) = 0,
*w*_{2}(T(\(\CP^n\))) = 0
for *n* odd, and = *x* for *n* even, where *x* is a generator of
H^{2}(\(\mathbb C\)P^{n};
\(\mathbb Z\)_{2}).

* *w*(TS^{n}) = 1
(i.e., same as for the trivial bundle).

* *w*(\(\gamma_n^{~1}\)) = 1 + *a*, where *a* is
a generator of H^{1}(\(\mathbb R\)P^{n};
\(\mathbb Z\)_{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* = \(w_1^~\)(T*M*)^{r1}
\(w_2^~\)(T*M*)^{r2} ···
\(w_n^~\)(T*M*)^{rn} ∈
H^{n}(*M*; \(\mathbb Z\)_{2}) ,

with *r*_{i} ≥ 0
and ∑_{i} *i*
*r*_{i} = *n*, is defined by

*w*_{1}^{r1}
*w*_{2}^{r2}
··· \(w_n^~\)^{rn} [*M*]:=
\(\langle\)*c*, *μ*_{M}\(\rangle\) ,

where *μ*_{M} 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