A
characteristic class
defined for real vector bundles (cf.
Vector bundle).
Pontryagin classes were introduced by
L.S. Pontryagin
[1]
in
1947.
For a vector bundle
with base
the Pontryagin classes are denoted by the symbol
and are defined to be equal to
,
where
is the complexification of
and
are the Chern classes (cf.
Chern class).
The
total Pontryagin class
is the non-homogeneous characteristic class
.
In other words, the Pontryagin classes are defined as homology classes
determined by the equality
,
where
is the mapping corresponding to the complexification of the universal vector bundle and
are the Chern classes.
Let
be the real bundle of the universal vector bundle
over
.
The total Pontryagin class
of the vector bundle
coincides with
,
where
are the
Wu generators
(see
Characteristic class).
A partial description of the cohomology ring
can be obtained in terms of Wu generators in the following way. The mapping
corresponding to the vector bundle
,
where
is the one-dimensional trivial vector bundle, induces a ring homomorphism
,
under which the subring of
generated by the Pontryagin classes
is mapped monomorphically onto the subring of
consisting of all even symmetric polynomials in the
Wu generators. Evenness is understood in the sense that the degree of every variable
in the polynomial should be even. Thus, an expression in
Wu generators is obtained for any element of the ring
.
This is important for practical calculations with Pontryagin classes. The characteristic
class determined by an even symmetric polynomial in the Wu
generators can be expressed in Pontryagin classes as follows. First,
the polynomial is written in elementary symmetric functions of the variables
and then the elementary symmetric functions are replaced by Pontryagin classes.
If
are two real vector bundles over a common base, then the cohomology class
is of order at most two; this is due to the fact that for the first Chern class
.
Let a ring
,
containing
,
be considered as the ring of coefficients, and let
be a Pontryagin class with values in
.
In this case the following equality is valid:
or
The ring
is monomorphically mapped into
and the image of this mapping coincides with the subring of
all even symmetric series in Wu generators as variables. Then
the total Pontryagin class is mapped to the polynomial
,
and the Pontryagin classes
— to elementary symmetric functions of
.
Theorem:
The cohomology ring
contains, beside a Pontryagin class, also the
Euler class
.
Theorem:
for the space
the equality
holds.
The mapping
can be extended to
.
The induced mapping
maps
to zero for
odd, and to
for
even.
Let
be a
formal power series
over the field
.
Then the series
determines some non-homogeneous element of the ring
,
i.e. a characteristic class. Admitting a certain freedom, one can write
The characteristic class
is
stable
(that is
,
where
is the trivial vector bundle) if and only if the constant term in
is equal to one. If one assumes
,
then the characteristic class constructed by the above-described method is denoted by
and is called the
Hirzebruch
-class,
The standard procedure of expressing the series
in elementary symmetric functions of
leads to the representation of
in the form of a series in Pontryagin classes. Another characteristic class
that is important for applications is obtained if it is assumed that
The class determined by the even symmetric series
is called the
-class.
Similarly, the
-class
is the characteristic class determined by the series
where
.
Both these classes, as well as
,
can be expressed in Pontryagin classes.
Topological invariance.
In
1965
S.P. Novikov
[2]
proved that the Pontryagin classes with rational coefficients coincide
for two homeomorphic manifolds. It was already known
that rational Pontryagin classes are piecewise-linearly invariant, i.e.
coincide for two piecewise-linear homeomorphic manifolds. Moreover,
rational Pontryagin classes were defined (see
[4])
for piecewise-linear manifolds (possibly with a boundary). An example was given (see
)
of integer Pontryagin classes which are not topological invariants.
In
1969
it was shown (see
[7])
that the fibre
of the bundle
has the homotopy type of the
Eilenberg–MacLane space
.
From this the
topological invariance
of rational Pontryagin classes follows, as well as a disproof of the
fundamental hypothesis of combinatorial topology
(the
Hauptvermutung).
Generalized Pontryagin classes.
Let
be a generalized cohomology theory (cf.
Generalized cohomology theories)
in which Chern classes
are defined. If for a one-dimensional complex vector bundle
the equality
holds, the Pontryagin classes with values in the theory
can be defined via the above-mentioned formula
.
The classes thus defined will have the property
,
where
is the total Pontryagin class considered in the theory
.
However, in many generalized theories used in practice (for example, in
-theory)
the equality proposed for
does not hold. In such theories it does not make sense to define
Pontryagin classes in the above-described manner, since under such a definition the usual
formula for the total class of the sum of two vector bundles, even after including
in the coefficients, is not valid. One can define
generalized Pontryagin classes in the following way. Let
be a multiplicative cohomology theory in which an orientation
of a vector bundle
,
where
is an arbitrary
-dimensional
real vector bundle over
,
is universally given. Let
be the Euler class of
,
,
where
is the inclusion of a zero section. Pontryagin classes in the theory
are the characteristic classes
,
defined for real vector bundles and satisfying the following conditions:
1)
if
;
2)
,
where
is the trivial bundle;
3)
is an element of order a power of two;
4)
,
where
.
The uniqueness and existence of characteristic classes with the above
properties has been proved. From this point of view,
Pontryagin classes lead to the notion of a two-valued
formal group
over the ring
corresponding to the theory
.
The characteristic classes
in
-theory
are defined by the following formula:
where
;
here
are the Chern classes in
-graded
-theory.