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 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 *w*_{1}(*γ*_{1}^{1}) ≠ 0,
where *γ*_{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**

* *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(\(\mathbb C\)P^{n}))
= 0, *w*_{2}(T(\(\mathbb C\)P^{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*(*γ*_{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.

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send feedback and suggestions to bombelli at olemiss.edu – modified 14
jan 2016