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P a g e s f o r l o g g e d o u t e d i t o r s l e a r n m o r e ● C o n t r i b u t i o n s ● T a l k
( T o p )
1 D e f i n i t i o n
2 P r o p e r t i e s
T o g g l e P r o p e r t i e s s u b s e c t i o n
2 . 1 P o n t r y a g i n c l a s s e s a n d c u r v a t u r e
2 . 2 P o n t r y a g i n c l a s s e s o f a m a n i f o l d
2 . 3 P o n t r y a g i n c l a s s e s f r o m C h e r n c l a s s e s
2 . 4 P o n t r y a g i n c l a s s e s o n a Q u a r t i c K 3 S u r f a c e
3 P o n t r y a g i n n u m b e r s
T o g g l e P o n t r y a g i n n u m b e r s s u b s e c t i o n
3 . 1 P r o p e r t i e s
4 G e n e r a l i z a t i o n s
5 S e e a l s o
6 R e f e r e n c e s
7 E x t e r n a l l i n k s
T o g g l e t h e t a b l e o f c o n t e n t s
P o n t r y a g i n c l a s s
8 l a n g u a g e s
● D e u t s c h ● F r a n ç a i s ● 한 국 어 ● 日 本 語 ● P o r t u g u ê s ● Р у с с к и й ● S v e n s k a ● У к р а ї н с ь к а
E d i t l i n k s
● A r t i c l e ● T a l k
E n g l i s h
● R e a d ● E d i t ● V i e w h i s t o r y
T o o l s
T o o l s
A c t i o n s
● R e a d ● E d i t ● V i e w h i s t o r y
G e n e r a l
● W h a t l i n k s h e r e ● R e l a t e d c h a n g e s ● U p l o a d f i l e ● S p e c i a l p a g e s ● P e r m a n e n t l i n k ● P a g e i n f o r m a t i o n ● C i t e t h i s p a g e ● G e t s h o r t e n e d U R L ● D o w n l o a d Q R c o d e ● W i k i d a t a i t e m
P r i n t / e x p o r t
● D o w n l o a d a s P D F ● P r i n t a b l e v e r s i o n
A p p e a r a n c e
F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
Given a real vector bundle
E
{\displaystyle E}
M
{\displaystyle M}
k
{\displaystyle k}
p
k
(
E )
{\displaystyle p_{k}(E )}
p
k
(
E )
=
p
k
(
E ,
Z
)
=
(
−
1
)
k
c
2 k
(
E ⊗
C
)
∈
H
4 k
(
M ,
Z
)
,
{\displaystyle p_{k}(E )=p_{k}(E,\mathbb {Z} )=(-1)^{k}c_{2k}(E\otimes \mathbb {C} )\in H^{4k}(M,\mathbb {Z} ),}
where:
c
2 k
(
E ⊗
C
)
{\displaystyle c_{2k}(E\otimes \mathbb {C} )}
2 k
{\displaystyle 2k}
Chern class of the complexification
E ⊗
C
=
E ⊕
i E
{\displaystyle E\otimes \mathbb {C} =E\oplus iE}
of
E
{\displaystyle E}
H
4 k
(
M ,
Z
)
{\displaystyle H^{4k}(M,\mathbb {Z} )}
4 k
{\displaystyle 4k}
cohomology group of
M
{\displaystyle M}
integer coefficients.
The rational Pontryagin class
p
k
(
E ,
Q
)
{\displaystyle p_{k}(E,\mathbb {Q} )}
p
k
(
E )
{\displaystyle p_{k}(E )}
in
H
4 k
(
M ,
Q
)
{\displaystyle H^{4k}(M,\mathbb {Q} )}
4 k
{\displaystyle 4k}
M
{\displaystyle M}
rational coefficients.
Properties [ edit ]
The total Pontryagin class
p (
E )
=
1 +
p
1
(
E )
+
p
2
(
E )
+
⋯
∈
H
∗
(
M ,
Z
)
,
{\displaystyle p(E )=1+p_{1}(E )+p_{2}(E )+\cdots \in H^{*}(M,\mathbb {Z} ),}
is (modulo 2-torsion) multiplicative with respect to
Whitney sum of vector bundles, i.e.,
2 p (
E ⊕
F )
=
2 p (
E )
⌣
p (
F )
{\displaystyle 2p(E\oplus F)=2p(E )\smile p(F )}
for two vector bundles
E
{\displaystyle E}
F
{\displaystyle F}
M
{\displaystyle M}
p
k
{\displaystyle p_{k}}
2
p
1
(
E ⊕
F )
=
2
p
1
(
E )
+
2
p
1
(
F )
,
{\displaystyle 2p_{1}(E\oplus F)=2p_{1}(E )+2p_{1}(F ),}
2
p
2
(
E ⊕
F )
=
2
p
2
(
E )
+
2
p
1
(
E )
⌣
p
1
(
F )
+
2
p
2
(
F )
{\displaystyle 2p_{2}(E\oplus 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
E
10
{\displaystyle E_{10}}
9-sphere . (The clutching function for
E
10
{\displaystyle E_{10}}
homotopy group
π
8
(
O
(
10 )
)
=
Z
/
2
Z
{\displaystyle \pi _{8}(\mathrm {O} (10 ))=\mathbb {Z} /2\mathbb {Z} }
w
9
{\displaystyle w_{9}}
of
E
10
{\displaystyle E_{10}}
Wu formula
w
9
=
w
1
w
8
+
S
q
1
(
w
8
)
{\displaystyle w_{9}=w_{1}w_{8}+Sq^{1}(w_{8})}
Whitney sum of
E
10
{\displaystyle E_{10}}
Hatcher 2009 , p. 76 )
Given a
2 k
{\displaystyle 2k}
E
{\displaystyle E}
p
k
(
E )
=
e (
E )
⌣
e (
E )
,
{\displaystyle p_{k}(E )=e(E )\smile e(E ),}
where
e (
E )
{\displaystyle e(E )}
Euler class of
E
{\displaystyle E}
⌣
{\displaystyle \smile }
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
p
k
(
E ,
Q
)
∈
H
4 k
(
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
{\displaystyle E}
n
{\displaystyle n}
differentiable manifold
M
{\displaystyle M}
connection , the total Pontryagin class is expressed as
p =
[
1 −
T r
(
Ω
2
)
8
π
2
+
T r
(
Ω
2
)
2
−
2
T r
(
Ω
4
)
128
π
4
−
T r
(
Ω
2
)
3
−
6
T r
(
Ω
2
)
T r
(
Ω
4
)
+
8
T r
(
Ω
6
)
3072
π
6
+
⋯
]
∈
H
d R
∗
(
M )
,
{\displaystyle 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}}}+\cdots \right]\in H_{dR}^{*}(M ),}
where
Ω
{\displaystyle \Omega }
curvature form , and
H
d R
∗
(
M )
{\displaystyle H_{dR}^{*}(M )}
de Rham cohomology groups.[1]
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} )}
If the dimension is at least five, there are at most finitely many different smooth manifolds with given homotopy type and Pontryagin classes.
Pontryagin classes from Chern classes [ edit ]
The Pontryagin classes of a complex vector bundle
π
:
E →
X
{\displaystyle \pi :E\to X}
E
⊗
R
C
≅
E ⊕
E ¯
{\displaystyle E\otimes _{\mathbb {R} }\mathbb {C} \cong E\oplus {\bar {E}}}
c
i
(
E ¯
)
=
(
−
1
)
i
c
i
(
E )
{\displaystyle c_{i}({\bar {E}})=(-1)^{i}c_{i}(E )}
c (
E ⊕
E ¯
)
=
c (
E )
c (
E ¯
)
{\displaystyle c(E\oplus {\bar {E}})=c(E )c({\bar {E}})}
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. On a surface, we have
(
1 −
c
1
(
E )
+
c
2
(
E )
)
(
1 +
c
1
(
E )
+
c
2
(
E )
)
=
1 −
c
1
(
E
)
2
+
2
c
2
(
E )
{\displaystyle (1-c_{1}(E )+c_{2}(E ))(1+c_{1}(E )+c_{2}(E ))=1-c_{1}(E )^{2}+2c_{2}(E )}
showing
p
1
(
E )
=
c
1
(
E
)
2
−
2
c
2
(
E )
{\displaystyle p_{1}(E )=c_{1}(E )^{2}-2c_{2}(E )}
c
2
(
L )
=
0
{\displaystyle c_{2}(L )=0}
Pontryagin classes on a Quartic K3 Surface [ edit ]
Recall that a quartic polynomial whose vanishing locus in
C P
3
{\displaystyle \mathbb {CP} ^{3}}
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}
c
2
(
X )
=
6 [
H
]
2
{\displaystyle c_{2}(X )=6[H ]^{2}}
[
H
]
2
{\displaystyle [H ]^{2}}
24
{\displaystyle 24}
p
1
(
X )
=
−
2
c
2
(
X )
{\displaystyle p_{1}(X )=-2c_{2}(X )}
p
1
(
X )
=
−
48
{\displaystyle p_{1}(X )=-48}
[3]
Pontryagin numbers [ edit ]
Pontryagin numbers are certain topological invariants of a smooth manifold . Each Pontryagin number of a manifold
M
{\displaystyle M}
M
{\displaystyle M}
manifold
M
{\displaystyle M}
Given a smooth
4 n
{\displaystyle 4n}
M
{\displaystyle M}
k
1
,
k
2
,
…
,
k
m
{\displaystyle k_{1},k_{2},\ldots ,k_{m}}
k
1
+
k
2
+
⋯
+
k
m
=
n
{\displaystyle k_{1}+k_{2}+\cdots +k_{m}=n}
the Pontryagin number
P
k
1
,
k
2
,
…
,
k
m
{\displaystyle P_{k_{1},k_{2},\dots ,k_{m}}}
P
k
1
,
k
2
,
…
,
k
m
=
p
k
1
⌣
p
k
2
⌣
⋯
⌣
p
k
m
(
[
M ]
)
{\displaystyle P_{k_{1},k_{2},\dots ,k_{m}}=p_{k_{1}}\smile p_{k_{2}}\smile \cdots \smile p_{k_{m}}([M ])}
where
p
k
{\displaystyle p_{k}}
k
{\displaystyle k}
[
M ]
{\displaystyle [M ]}
fundamental class of
M
{\displaystyle M}
Properties [ edit ]
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}}}
Hirzebruch signature theorem .
Generalizations [ edit ]
There is also a quaternionic Pontryagin class, for vector bundles with quaternion structure.
See also [ edit ]
References [ edit ]
^ "A Survey of Computations of Homotopy Groups of Spheres and Cobordisms" (PDF) . p. 16. Archived (PDF) from the original on 2016-01-22.
External links [ edit ]
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Pontryagin_class&oldid=1232953301 " C a t e g o r i e s : ● C h a r a c t e r i s t i c c l a s s e s ● D i f f e r e n t i a l t o p o l o g y
● T h i s p a g e w a s l a s t e d i t e d o n 6 J u l y 2 0 2 4 , a t 1 4 : 2 3 ( U T C ) . ● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n . ● P r i v a c y p o l i c y ● A b o u t W i k i p e d i a ● D i s c l a i m e r s ● C o n t a c t W i k i p e d i a ● C o d e o f C o n d u c t ● D e v e l o p e r s ● S t a t i s t i c s ● C o o k i e s t a t e m e n t ● M o b i l e v i e w
denotes the 
-th Chern class of the complexification 
of
is the 
-cohomology group of 
![{\displaystyle 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}}}+\cdots \right]\in H_{dR}^{*}(M),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/398b2d853e577a27bd8eefa3fc8264df98c8a61a)
![{\displaystyle c_{2}(X)=6[H]^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a7521387bcea6b649cee551172417c4298ee2e6)
![{\displaystyle [H]^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46055fd0a533d44b58327dbeb0087ecbdd27913e)
such that 
,
![{\displaystyle P_{k_{1},k_{2},\dots ,k_{m}}=p_{k_{1}}\smile p_{k_{2}}\smile \cdots \smile p_{k_{m}}([M])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/176640f7fd7c32d30f974a94992542b49fcb4b7c)
![{\displaystyle [M]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5ca74e595b2281c0aef1897ecafa282d1f182e2)
-genus can be expressed through Pontryagin numbers. For the theorem describing the linear combination of Pontryagin numbers giving the signature see Hirzebruch signature theorem.
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29142a0afed50bb796bc43e205bffd1c70672393)