Pontryagin class
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:
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or
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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:
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The cohomology ring contains, beside a Pontryagin class, also the Euler class
. Theorem:
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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
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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,
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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
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The class determined by the even symmetric series
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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:
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where ; here
are the Chern classes in
-graded
-theory.
References
[1] | L.S. Pontryagin, "Characteristic classes of differentiable manifolds" Transl. Amer. Math. Soc. (1) , 7 (1962) pp. 279–331 Mat. Sb. , 21 (1947) pp. 233–284 |
[2] | S.P. Novikov, "Topological invariance of rational Pontrjagin classes" Soviet Math. Dokl. , 6 : 4 (1965) pp. 921–923 Dokl. Akad. Nauk SSSR , 163 (1965) pp. 298–300 Zbl 0146.19503 Zbl 0146.19502 |
[3] | V.M. Bukhshtaber, "The Chern–Dold character in cobordisms. I" Math. USSR Sb. , 12 : 4 (1970) pp. 573–594 Mat. Sb. , 83 (1970) pp. 575–595 Zbl 0219.57027 |
[4] | V.A. Rokhlin, A.S. Shvarč, "The combinatorial invariance of Pontrjagin classes" Dokl. Akad. Nauk SSSR , 114 (1957) pp. 490–493 (In Russian) Zbl 0078.36803 |
[5a] | J. Milnor, Matematika , 3 : 4 (1959) pp. 3–53 MR0339964 |
[5b] | J. Milnor, Matematika , 9 : 4 (1965) pp. 3–40 MR0339964 |
[6] | R.E. Stong, "Notes on cobordism theory" , Princeton Univ. Press (1968) MR0248858 Zbl 0181.26604 |
[7] | R.C. Kirby, L.C. Siebenmann, "Foundational essays on topological manifolds, smoothings, and triangulations" , Princeton Univ. Press (1977) MR0645390 Zbl 0361.57004 |
Comments
References
[a1] | J.W. Milnor, J.D. Stasheff, "Characteristic classes" , Princeton Univ. Press (1974) MR0440554 Zbl 0298.57008 |
[a2] | D.W. Anderson, E.H. Brown, F.P. Peterson, "Spin cobordism" Ann. of Math. (To appear) MR0219077 MR0190939 Zbl 0156.21605 Zbl 0136.44103 |
[a3] | J. Dieudonné, "A history of algebraic and differential topology 1900–1960" , Birkhäuser (1989) MR0995842 Zbl 0673.55002 |
Pontryagin class. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pontryagin_class&oldid=48239