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Pontryagin class
In mathematics, the Pontryagin classes, named for Lev Pontryagin, are certain characteristic classes. The Pontryagin class lies in cohomology groups with degree a multiple of four. It applies to real vector bundles.
Contents
1 Definition
2 Properties
2.1 Pontryagin classes and curvature
2.2 Pontryagin classes of a manifold
3 Pontryagin numbers
3.1 Properties
4 Generalizations
5 See also
6 References
7 External links
Definition
Given a real vector bundle E over M, its k-th Pontryagin class pk(E) is defined as
pk(E) = pk(E, Z) = (−1)kc2k(E ⊗ C) ∈ H4k(M, Z),
where:
c2k(E ⊗ C) denotes the 2k-th Chern class of the complexification E ⊗ C = E ⊕ iE of E,
H4k(M, Z) is the 4k-cohomology group of M with integer coefficients.
The rational Pontryagin class pk(E, Q) is defined to be the image of pk(E) in H4k(M, Q), the 4k-cohomology group of M with rational coefficients.
Properties
The total Pontryagin class
- p(E)=1+p1(E)+p2(E)+⋯∈H∗(M,Z),{displaystyle p(E)=1+p_{1}(E)+p_{2}(E)+cdots in H^{*}(M,mathbf {Z} ),}
is (modulo 2-torsion) multiplicative with respect to
Whitney sum of vector bundles, i.e.,
- 2p(E⊕F)=2p(E)⌣p(F){displaystyle 2p(Eoplus F)=2p(E)smile p(F)}
for two vector bundles E and F over M. In terms of the individual Pontryagin classes pk,
- 2p1(E⊕F)=2p1(E)+2p1(F),{displaystyle 2p_{1}(Eoplus F)=2p_{1}(E)+2p_{1}(F),}
- 2p2(E⊕F)=2p2(E)+2p1(E)⌣p1(F)+2p2(F){displaystyle 2p_{2}(Eoplus F)=2p_{2}(E)+2p_{1}(E)smile p_{1}(F)+2p_{2}(F)}
and so on.
The vanishing of the Pontryagin classes and Stiefel–Whitney classes of a vector bundle does not guarantee that the vector bundle is trivial. For example, up to vector bundle isomorphism, there is a unique nontrivial rank 10 vector bundle E10 over the 9-sphere. (The clutching function for E10 arises from the homotopy group π8(O(10)) = Z/2Z.) The Pontryagin classes and Stiefel-Whitney classes all vanish: the Pontryagin classes don't exist in degree 9, and the Stiefel–Whitney class w9 of E10 vanishes by the Wu formula w9 = w1w8 + Sq1(w8). Moreover, this vector bundle is stably nontrivial, i.e. the Whitney sum of E10 with any trivial bundle remains nontrivial. (Hatcher 2009, p. 76)
Given a 2k-dimensional vector bundle E we have
- pk(E)=e(E)⌣e(E),{displaystyle p_{k}(E)=e(E)smile e(E),}
where e(E) denotes the Euler class of E, and ⌣{displaystyle smile }
Pontryagin classes and curvature
As was shown by Shiing-Shen Chern and André Weil around 1948, the rational Pontryagin classes
- pk(E,Q)∈H4k(M,Q){displaystyle p_{k}(E,mathbf {Q} )in H^{4k}(M,mathbf {Q} )}
can be presented as differential forms which depend polynomially on the curvature form of a vector bundle. This Chern–Weil theory revealed a major connection between algebraic topology and global differential geometry.
For a vector bundle E over a n-dimensional differentiable manifold M equipped with a connection, the total Pontryagin class is expressed as
- p=[1−Tr(Ω2)8π2+Tr(Ω2)2−2Tr(Ω4)128π4−Tr(Ω2)3−6Tr(Ω2)Tr(Ω4)+8Tr(Ω6)3072π6+⋯]∈HdR∗(M),{displaystyle p=left[1-{frac {{rm {Tr}}(Omega ^{2})}{8pi ^{2}}}+{frac {{rm {Tr}}(Omega ^{2})^{2}-2{rm {Tr}}(Omega ^{4})}{128pi ^{4}}}-{frac {{rm {Tr}}(Omega ^{2})^{3}-6{rm {Tr}}(Omega ^{2}){rm {Tr}}(Omega ^{4})+8{rm {Tr}}(Omega ^{6})}{3072pi ^{6}}}+cdots right]in H_{dR}^{*}(M),}
![p=left[1-frac{{rm Tr}(Omega ^2)}{8 pi ^2}+frac{{rm Tr}(Omega ^2)^2-2 {rm Tr}(Omega ^4)}{128 pi ^4}-frac{{rm Tr}(Omega ^2)^3-6 {rm Tr}(Omega ^2) {rm Tr}(Omega ^4)+8 {rm Tr}(Omega ^6)}{3072 pi ^6}+cdotsright]in H^*_{dR}(M),](https://wikimedia.org/api/rest_v1/media/math/render/svg/398b2d853e577a27bd8eefa3fc8264df98c8a61a)
where Ω denotes the curvature form, and H*dR(M) denotes the de Rham cohomology groups.[citation needed]
Pontryagin classes of a manifold
The Pontryagin classes of a smooth manifold are defined to be the Pontryagin classes of its tangent bundle.
Novikov proved in 1966 that if manifolds are homeomorphic then their rational Pontryagin classes pk(M, Q) in H4k(M, Q) are the same.
If the dimension is at least five, there are at most finitely many different smooth manifolds with given homotopy type and Pontryagin classes.
Pontryagin numbers
Pontryagin numbers are certain topological invariants of a smooth manifold. The Pontryagin number vanishes if the dimension of manifold is not divisible by 4. It is defined in terms of the Pontryagin classes of a manifold as follows:
Given a smooth 4n{displaystyle 4n}
k1,k2,…,km{displaystyle k_{1},k_{2},ldots ,k_{m}}such that k1+k2+⋯+km=n{displaystyle k_{1}+k_{2}+cdots +k_{m}=n}
,
the Pontryagin number Pk1,k2,…,km{displaystyle P_{k_{1},k_{2},dots ,k_{m}}}
- Pk1,k2,…,km=pk1⌣pk2⌣⋯⌣pkm([M]){displaystyle P_{k_{1},k_{2},dots ,k_{m}}=p_{k_{1}}smile p_{k_{2}}smile cdots smile p_{k_{m}}([M])}
![P_{k_1,k_2,dots, k_m}=p_{k_1}smile p_{k_2}smile cdotssmile p_{k_m}([M])](https://wikimedia.org/api/rest_v1/media/math/render/svg/176640f7fd7c32d30f974a94992542b49fcb4b7c)
where pk{displaystyle p_{k}}
Properties
- Pontryagin numbers are oriented cobordism invariant; and together with Stiefel-Whitney numbers they determine an oriented manifold's oriented cobordism class.
- Pontryagin numbers of closed Riemannian manifolds (as well as Pontryagin classes) can be calculated as integrals of certain polynomials from the curvature tensor of a Riemannian manifold.
- Invariants such as signature and A^{displaystyle {hat {A}}}
-genus can be expressed through Pontryagin numbers. For the theorem describing the linear combination of Pontryagin numbers giving the signature see Hirzebruch signature theorem.
Generalizations
There is also a quaternionic Pontryagin class, for vector bundles with quaternion structure.
See also
- Chern–Simons form
References
Milnor John W.; Stasheff, James D. (1974). Characteristic classes. Annals of Mathematics Studies. Princeton, New Jersey; Tokyo: Princeton University Press / University of Tokyo Press. ISBN 0-691-08122-0..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
Hatcher, Allen (2009). "Vector Bundles & K-Theory" (2.1 ed.).
External links
Hazewinkel, Michiel, ed. (2001) [1994], "Pontryagin class", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
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