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In mathematics, the Pontryagin classes, named after Lev Pontryagin, are certain characteristic classes of real vector bundles. The Pontryagin classes lie in cohomology groups with degrees a multiple of four.
Given a real vector bundle E {\displaystyle E} over M {\displaystyle M} , its k {\displaystyle k} -th Pontryagin class p k ( E ) {\displaystyle p_{k}(E)} is defined as
where:
The rational Pontryagin class p k ( E , Q ) {\displaystyle p_{k}(E,\mathbb {Q} )} is defined to be the image of p k ( E ) {\displaystyle p_{k}(E)} in H 4 k ( M , Q ) {\displaystyle H^{4k}(M,\mathbb {Q} )} , the 4 k {\displaystyle 4k} -cohomology group of M {\displaystyle M} with rational coefficients.
The total Pontryagin class
is (modulo 2-torsion) multiplicative with respect to Whitney sum of vector bundles, i.e.,
for two vector bundles E {\displaystyle E} and F {\displaystyle F} over M {\displaystyle M} . In terms of the individual Pontryagin classes p k {\displaystyle p_{k}} ,
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 E 10 {\displaystyle E_{10}} over the 9-sphere. (The clutching function for E 10 {\displaystyle E_{10}} arises from the homotopy group π 8 ( O ( 10 ) ) = Z / 2 Z {\displaystyle \pi _{8}(\mathrm {O} (10))=\mathbb {Z} /2\mathbb {Z} } .) The Pontryagin classes and Stiefel-Whitney classes all vanish: the Pontryagin classes don't exist in degree 9, and the Stiefel–Whitney class w 9 {\displaystyle w_{9}} of E 10 {\displaystyle E_{10}} vanishes by the Wu formula w 9 = w 1 w 8 + S q 1 ( w 8 ) {\displaystyle w_{9}=w_{1}w_{8}+Sq^{1}(w_{8})} . Moreover, this vector bundle is stably nontrivial, i.e. the Whitney sum of E 10 {\displaystyle E_{10}} with any trivial bundle remains nontrivial. (Hatcher 2009, p. 76)
Given a 2 k {\displaystyle 2k} -dimensional vector bundle E {\displaystyle E} we have
where e ( E ) {\displaystyle e(E)} denotes the Euler class of E {\displaystyle E} , and ⌣ {\displaystyle \smile } denotes the cup product of cohomology classes.
Pontryagin classes and curvature[edit]As was shown by Shiing-Shen Chern and André Weil around 1948, the rational Pontryagin classes
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 {\displaystyle E} over a n {\displaystyle n} -dimensional differentiable manifold M {\displaystyle M} equipped with a connection, the total Pontryagin class is expressed as
where Ω {\displaystyle \Omega } denotes the curvature form, and H d R ∗ ( M ) {\displaystyle H_{dR}^{*}(M)} denotes the de Rham cohomology groups.[citation needed]
Pontryagin classes of a manifold[edit]The Pontryagin classes of a smooth manifold are defined to be the Pontryagin classes of its tangent bundle.
Novikov proved in 1966 that if two compact, oriented, smooth manifolds are homeomorphic then their rational Pontryagin classes p k ( M , Q ) {\displaystyle p_{k}(M,\mathbf {Q} )} in H 4 k ( M , Q ) {\displaystyle H^{4k}(M,\mathbf {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.[1]
Pontryagin classes from Chern classes[edit]The Pontryagin classes of a complex vector bundle π : E → X {\displaystyle \pi :E\to X} is completely determined by its Chern classes. This follows from the fact that E ⊗ R C ≅ E ⊕ E ¯ {\displaystyle E\otimes _{\mathbb {R} }\mathbb {C} \cong E\oplus {\bar {E}}} , the Whitney sum formula, and properties of Chern classes of its complex conjugate bundle. That is, c i ( E ¯ ) = ( − 1 ) i c i ( E ) {\displaystyle c_{i}({\bar {E}})=(-1)^{i}c_{i}(E)} and c ( E ⊕ E ¯ ) = c ( E ) c ( E ¯ ) {\displaystyle c(E\oplus {\bar {E}})=c(E)c({\bar {E}})} . Then, given this relation, we can see
1 − p 1 ( E ) + p 2 ( E ) − ⋯ + ( − 1 ) n p n ( E ) = ( 1 + c 1 ( E ) + ⋯ + c n ( E ) ) ⋅ ( 1 − c 1 ( E ) + c 2 ( E ) − ⋯ + ( − 1 ) n c n ( E ) ) {\displaystyle 1-p_{1}(E)+p_{2}(E)-\cdots +(-1)^{n}p_{n}(E)=(1+c_{1}(E)+\cdots +c_{n}(E))\cdot (1-c_{1}(E)+c_{2}(E)-\cdots +(-1)^{n}c_{n}(E))} [2].
For example, we can apply this formula to find the Pontryagin classes of a complex vector bundle on a curve and a surface. For a curve, we have
( 1 − c 1 ( E ) ) ( 1 + c 1 ( E ) ) = 1 + c 1 ( E ) 2 {\displaystyle (1-c_{1}(E))(1+c_{1}(E))=1+c_{1}(E)^{2}}
so all of the Pontryagin classes of complex vector bundles are trivial. In general, looking at first two terms of the product
( 1 − c 1 ( E ) + c 2 ( E ) + … + ( − 1 ) n c n ( E ) ) ( 1 + c 1 ( E ) + c 2 ( E ) + … + c n ( E ) ) = 1 − c 1 ( E ) 2 + 2 c 2 ( E ) + … {\displaystyle (1-c_{1}(E)+c_{2}(E)+\ldots +(-1)^{n}c_{n}(E))(1+c_{1}(E)+c_{2}(E)+\ldots +c_{n}(E))=1-c_{1}(E)^{2}+2c_{2}(E)+\ldots }
we can see that p 1 ( E ) = c 1 ( E ) 2 − 2 c 2 ( E ) {\displaystyle p_{1}(E)=c_{1}(E)^{2}-2c_{2}(E)} . In particular, for line bundles this simplifies further since c 2 ( L ) = 0 {\displaystyle c_{2}(L)=0} by dimension reasons.
Pontryagin classes on a Quartic K3 Surface[edit]Recall that a quartic polynomial whose vanishing locus in C P 3 {\displaystyle \mathbb {CP} ^{3}} is a smooth subvariety is a K3 surface. If we use the normal sequence
0 → T X → T C P 3 | X → O ( 4 ) → 0 {\displaystyle 0\to {\mathcal {T}}_{X}\to {\mathcal {T}}_{\mathbb {CP} ^{3}}|_{X}\to {\mathcal {O}}(4)\to 0}
we can find
c ( T X ) = c ( T C P 3 | X ) c ( O ( 4 ) ) = ( 1 + [ H ] ) 4 ( 1 + 4 [ H ] ) = ( 1 + 4 [ H ] + 6 [ H ] 2 ) ⋅ ( 1 − 4 [ H ] + 16 [ H ] 2 ) = 1 + 6 [ H ] 2 {\displaystyle {\begin{aligned}c({\mathcal {T}}_{X})&={\frac {c({\mathcal {T}}_{\mathbb {CP} ^{3}}|_{X})}{c({\mathcal {O}}(4))}}\\&={\frac {(1+[H])^{4}}{(1+4[H])}}\\&=(1+4[H]+6[H]^{2})\cdot (1-4[H]+16[H]^{2})\\&=1+6[H]^{2}\end{aligned}}}
showing c 1 ( X ) = 0 {\displaystyle c_{1}(X)=0} and c 2 ( X ) = 6 [ H ] 2 {\displaystyle c_{2}(X)=6[H]^{2}} . Since [ H ] 2 {\displaystyle [H]^{2}} corresponds to four points, due to Bézout's lemma, we have the second chern number as 24 {\displaystyle 24} . Since p 1 ( X ) = − 2 c 2 ( X ) {\displaystyle p_{1}(X)=-2c_{2}(X)} in this case, we have
p 1 ( X ) = − 48 {\displaystyle p_{1}(X)=-48} . This number can be used to compute the third stable homotopy group of spheres.[3]
Pontryagin numbers[edit]Pontryagin numbers are certain topological invariants of a smooth manifold. Each Pontryagin number of a manifold M {\displaystyle M} vanishes if the dimension of M {\displaystyle M} is not divisible by 4. It is defined in terms of the Pontryagin classes of the manifold M {\displaystyle M} as follows:
Given a smooth 4 n {\displaystyle 4n} -dimensional manifold M {\displaystyle M} and a collection of natural numbers
the Pontryagin number P k 1 , k 2 , … , k m {\displaystyle P_{k_{1},k_{2},\dots ,k_{m}}} is defined by
where p k {\displaystyle p_{k}} denotes the k {\displaystyle k} -th Pontryagin class and [ M ] {\displaystyle [M]} the fundamental class of M {\displaystyle M} .
There is also a quaternionic Pontryagin class, for vector bundles with quaternion structure.
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