nLab
Pontryagin class

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

The Pontryagin classes are characteristic classes on the classifying space BO(n)B O(n) of the orthogonal group and, by pullback, on the base of any bundle with structural group the orthogonal group. The latter is where they were originally defined.

The analogs for the unitary group are the Chern classes.

Definition

The universal Pontryagin characteristic classes P kP_k on the classifying space BO(n)B O(n) are, up to a sign, the pullbacks of the Chern classes c 2kc_{2k} along the complexification inclusion

BO(n)BU(n). B O(n) \to B U(n) \,.

Properties

As generating universal characteristic classes

The torsion-free quotient of the cohomology ring H (BSO(2n+1),)H^\bullet(B SO(2n+1), \mathbb{Z}) is the polynomial ring on all Pontryagin classes {P i} i=1 n\{P_i\}_{i = 1}^n. The torsion is generated by Bockstein images of H (BSO(2n+1);𝔽 2)H^\bullet(BSO(2n+1);\mathbb{F}_2), which is generated by the Stiefel-Whitney classes.

The torsion-free quotient of the cohomology ring H (BSO(2n),)H^\bullet(B SO(2n), \mathbb{Z}) is the quotient of the polynomial ring on Pontryagin classes P iP_i and the Euler class χ\chi by the relation χ 2=P n\chi^2 = P_n; again the torsion is generated by Bocksteins of monomials in the Stiefel–Whitney classes.

Further relation to Chern classes

Under the other canonical map

j:BU(n)BO(2n) j \;\colon\; B U(n) \to BO(2n)

one has

j *(P k)= a+b=2k(1) a+kc ac b j^\ast(P_k) = \sum_{a + b = 2 k} (-1)^{a+k} c_a c_b

and

j *(χ)=c n. j^\ast(\chi) = c_n \,.

Splitting principle and Chern roots

Under the inclusion

i:U(1) nU(n)O(2n) i \;\colon\; U(1)^n \hookrightarrow U(n) \to O(2n)

of the maximal torus one has that

(Bi) *(P k)=σ k(x 1,,x n) 2 (B i)^\ast(P_k) = \sigma_k(x_1, \cdots, x_n)^2

and

(Bi) *(χ)=σ n(x 1,,x n) (B i)^\ast(\chi) = \sigma_n(x_1, \cdots, x_n)

where the x iH (BU(1) n,)x_i \in H^\bullet(B U(1)^n, \mathbb{Z}) are the “Chern roots”.

See at Chern class - Properties – Splitting principle and Chern roots and at splitting principle - Examples - Real vector bundles for more.

Chern-, Pontrjagin-, and Euler- characteristic forms

We spell out the formulas for the images under the Chern-Weil homomorphism of the Chern classes, Pontrjagin classes and Euler classes as characteristic forms over smooth manifolds.

Preliminaries

Let XX be a smooth manifold.

Write

(1)Ω 2(X)CAlg \Omega^{2\bullet}(X) \;\; \in \; CAlg_{\mathbb{R}}

for the commutative algebra over the real numbers of even-degree differential forms on XX, under the wedge product of differential forms. This is naturally a graded commutative algebra, graded by form degree, but since we consider only forms in even degree it is actually a plain commutative algebra, too, after forgetting the grading.

Let 𝔤\mathfrak{g} be a semisimple Lie algebra (such as 𝔰𝔲 ( d ) \mathfrak{su}(d) or 𝔰𝔬 ( d ) \mathfrak{so}(d) ) with Lie algebra representation VRep (𝔤)V \,\in\, Rep_{\mathbb{C}}(\mathfrak{g}) over the complex numbers of finite dimension dim (V)=ndim_{\mathbb{C}}(V) \,=\, n \,\in\, \mathbb{N} (for instance the adjoint representation or the fundamental representation), hence a homomorphism of Lie algebras

𝔤ρEnd (V) \mathfrak{g} \xrightarrow{\;\;\rho\;\;} End_{\mathbb{C}}(V)

to the linear endomorphism ring End (V)End_{\mathbb{C}}(V), regarded here through its commutator as the endomorphism Lie algebra of VV.

When regarded as an associative ring this is isomorphic to the matrix algebra of n×nn \times n square matrices

(2)End (V)Mat n×n(). End_{\mathbb{C}}(V) \;\; \simeq \;\; Mat_{n \times n}(\mathbb{C}) \,.

The tensor product of the \mathbb{C}-algebras (1) and (2)

is equivalently the n×nn \times n matrix algebra with coefficients in the complexification of even-degree differential forms:

Ω 2(X) End (V)Ω 2(X) (Mat n×n())Mat n×n(Ω 2(X) ). \Omega^{2\bullet} \big(X\big) \otimes_{\mathbb{R}} End_{\mathbb{C}}(V) \;\simeq\; \Omega^{2\bullet}(X) \otimes_{\mathbb{R}} \big( Mat_{n \times n}( \mathbb{R} ) \big) \;\; \simeq \;\; Mat_{n \times n} \big( \Omega^{2\bullet}(X) \otimes_{\mathbb{R}} \mathbb{C} \big) \,.

The multiplicative unit

(3)IMat n×n(Ω 2(X) ) I \;\in\; Mat_{n \times n} \big( \Omega^{2\bullet}(X) \otimes_{\mathbb{R}} \mathbb{C} \big)

in this algebra is the smooth function (differential 0-forms) which is constant on the n×nn \times n identity matrix and independent of tt.

Given a connection on a GG-principal bundle, we regard its 𝔤 \mathfrak{g} -valued curvature form as an element of this algebra

(4)F Ω 2(X) 𝔤ρΩ 2(X) End (V)Ω 2(X) End (V)[t]Mat n×n( Ω 2(X)). F_\nabla \,\in\, \Omega^2(X) \otimes_{\mathbb{R}} \mathfrak{g} \xrightarrow{\; \rho \;} \Omega^2(X) \otimes_{\mathbb{R}} End_{\mathbb{C}}(V) \xhookrightarrow{\;\;\;} \Omega^{2\bullet}(X) \otimes_{\mathbb{R}} End_{\mathbb{C}}(V)[t] \;\simeq\; Mat_{n \times n} \Big( \mathbb{C} \otimes_{\mathbb{R}} \Omega^{2}(X) \Big) \,.

The formulas

Chern forms

The total Chern form c()c(\nabla) is the determinant of the sum of the unit (3) with the curvature form (4), and its component in degree 2k2k, for kk \in \mathbb{N}, is the kkth Chern form c k()c_k(\nabla):

c() kc k()deg=2kdet(I+tiF 2π). c(\nabla) \;\; \coloneqq \;\; \sum_k \underset{ \mathclap{ deg = 2k } }{ \underbrace{ c_k(\nabla) } } \;\; \coloneqq \;\; det \left( I + t \frac{i F_\nabla}{2\pi} \right) \,.

By the relation between determinant and trace, this is equal to the exponential of the trace of the logarithm of I+iF 2πI + \frac{i F_\nabla}{2\pi}, this being the exponential series in the trace of the Mercator series in iF 2π\frac{i F_\nabla}{2\pi}:

(5)c() =det(I+tiF 2π) =exptrln(I+iF 2π) =exptr(k +1k(F 2πi) k) =exp(k +1k((i) k(2π) ktr(F k))) =1 =+11(itr(F )2π+12tr((F ) 2)(2π) 2i13tr((F ) 3)(2π) 314tr((F ) 4)(2π) 4+) =+12(itr(F )2π+12tr((F ) 2)(2π) 2i13tr((F ) 3)(2π) 314tr((F ) 4)(2π) 4+) 2 =+16(itr(F )2π+12tr((F ) 2)(2π) 2i13tr((F ) 3)(2π) 314tr((F ) 4)(2π) 4+) 3 =+124(itr(F )2π+12tr((F ) 2)(2π) 2i13tr((F ) 3)(2π) 314tr((F ) 4)(2π) 4+) 4 =+ =1 =+itr(F )2π =+12tr((F ) 2)(2π) 2+12(itr(F )2π) 2 =i13tr((F ) 3)(2π) 3+12(2(itr(F )2π)(12tr((F ) 2)(2π) 2))+16((itr(F )2π) 3) =14tr((F ) 4)(2π) 4+12(12tr((F ) 2)(2π) 2) 2+124(itr(F )2π) 4 =+ =1 =+itr(F )2π=c 1() =+tr((F ) 2)(tr(F )) 28π 2=c 2() =+i2tr((F ) 3)+3tr(F )tr((F ) 2)(tr(F )) 348π 3=c 3() =+6tr((F ) 4)+3tr((F ) 2) 2+(tr(F )) 4384π 4=c 4() =+ \begin{aligned} c(\nabla) & \;=\; det \left( I + t \frac{i F_\nabla}{2\pi} \right) \\ & \;=\; \exp \circ tr \circ ln \left( I + \frac{i F_\nabla}{2\pi} \right) \\ & \;=\; \exp \circ tr \left( - \underset {k \in \mathbb{N}_+} {\sum} \tfrac{1}{k} \left( \frac{F_\nabla}{2\pi i} \right)^k \right) \\ & \;=\; \exp \left( \underset {k \in \mathbb{N}_+} {\sum} \tfrac{1}{k} \left( \frac { - (-i)^k } {(2\pi)^k} tr\big( F_\nabla^{\wedge_k} \big) \right) \right) \\ & \;=\; 1 \\ & \phantom{\;=\;} + \phantom{\frac{1}{1}} \left( i \tfrac{ tr\big(F_\nabla\big) }{2 \pi} + \tfrac{1}{2} \tfrac{ tr\big( (F_\nabla)^{2} \big)}{(2 \pi)^2} -i \tfrac{1}{3} \tfrac{ tr\big( (F_\nabla)^{3} \big)}{(2 \pi)^3} - \tfrac{1}{4} \tfrac{ tr\big( (F_\nabla)^{4} \big)}{(2 \pi)^4} + \cdots \right) \\ & \phantom{\;=\;} + \frac{1}{2} \left( i \tfrac{ tr\big(F_\nabla\big) }{2 \pi} + \tfrac{1}{2} \tfrac{ tr\big( (F_\nabla)^{2} \big)}{(2 \pi)^2} -i \tfrac{1}{3} \tfrac{ tr\big( (F_\nabla)^{3} \big)}{(2 \pi)^3} - \tfrac{1}{4} \tfrac{ tr\big( (F_\nabla)^{4} \big)}{(2 \pi)^4} + \cdots \right)^2 \\ & \phantom{\;=\;} + \frac{1}{6} \left( i \tfrac{ tr\big(F_\nabla\big) }{2 \pi} + \tfrac{1}{2} \tfrac{ tr\big( (F_\nabla)^{2} \big)}{(2 \pi)^2} -i \tfrac{1}{3} \tfrac{ tr\big( (F_\nabla)^{3} \big)}{(2 \pi)^3} - \tfrac{1}{4} \tfrac{ tr\big( (F_\nabla)^{4} \big)}{(2 \pi)^4} + \cdots \right)^3 \\ & \phantom{\;=\;} + \frac{1}{24} \left( i \tfrac{ tr\big(F_\nabla\big) }{2 \pi} + \tfrac{1}{2} \tfrac{ tr\big( (F_\nabla)^{2} \big)}{(2 \pi)^2} -i \tfrac{1}{3} \tfrac{ tr\big( (F_\nabla)^{3} \big)}{(2 \pi)^3} - \tfrac{1}{4} \tfrac{ tr\big( (F_\nabla)^{4} \big)}{(2 \pi)^4} + \cdots \right)^4 \\ & \phantom{\;=\;} + \cdots \\ & \;=\; 1 \\ & \phantom{\;=\;} + i \frac { tr\big(F_\nabla\big) } { 2 \pi } \\ & \phantom{\;=\;} + \tfrac{1}{2} \frac { tr\big( (F_\nabla)^2 \big) } { (2 \pi)^2 } + \frac{1}{2} \left( i \frac { tr\big( F_\nabla \big) } { 2\pi } \right)^2 \\ & \phantom{\;=\;} - i \tfrac{1}{3} \frac { tr\big( (F_\nabla)^3 \big) } { (2 \pi)^3 } + \frac{1}{2} \left( 2 \left( i \frac { tr\big( F_\nabla \big) } { 2 \pi } \right) \left( \tfrac{1}{2} \frac { tr\big( (F_\nabla)^2 \big) } { (2 \pi)^2 } \right) \right) + \frac{1}{6} \left( \left( i \frac { tr\big(F_\nabla\big) } { 2\pi } \right)^3 \right) \\ & \phantom{\;=\;} - \tfrac{1}{4} \frac {tr\big( (F_\nabla)^4 \big)} { (2 \pi)^4 } + \frac{1}{2} \left( \tfrac{1}{2} \frac {tr\big( (F_\nabla)^2 \big)} { (2 \pi)^2 } \right)^2 + \frac{1}{24} \left( i \frac {tr\big( F_\nabla \big)} { 2\pi } \right)^4 \\ & \phantom{\;=\;} + \cdots \\ & \;=\; 1 \\ & \phantom{\;=\;} + \underset{ \color{blue} = c_1(\nabla) }{ \underbrace{ i \frac { tr\big(F_\nabla\big) } { 2 \pi } }} \\ & \phantom{\;=\;} + \underset{ \color{blue} = c_2(\nabla) }{ \underbrace{ \frac {\tr\big( (F_\nabla)^2 \big) - \big( tr(F_\nabla) \big)^2 } { 8 \pi^2 } }} \\ & \phantom{\;=\;} + \underset{ \color{blue} = c_3(\nabla) }{ \underbrace{ i \frac { - 2 \cdot tr\big( (F_\nabla)^3 \big) + 3 \cdot tr(F_\nabla) \cdot tr\big( (F_\nabla)^2 \big) - \big( tr(F_\nabla ) \big)^3 } {48 \pi^3} }} \\ & \phantom{\;=\;} + \underset{ \color{blue} = c_4(\nabla) }{ \underbrace{ \frac { -6 \cdot tr\big( (F_\nabla)^4 \big) + 3 \cdot tr\big( (F_\nabla)^2 \big)^2 + \big( tr(F_\nabla) \big)^4 } {384 \pi^4} }} \\ & \phantom{\;=\;} + \cdots \end{aligned}
Pontrjagin forms

Setting tr(F )=0tr(F_\nabla) = 0 in these expressions (5) yields the total Pontrjagin form p()p(\nabla) with degree=4k4k-components the Pontrjagin forms p k()p_{k}(\nabla):

p() k(1) kp k()deg=4k =kc 2k()deg=4k =1 =+tr((F ) 2)8π 2=p 1() =+2tr((F ) 4)+tr((F ) 2) 2128π 4=p 2() =+ \begin{aligned} p(\nabla) & \;\coloneqq\; \underset{k \in \mathbb{N}}{\sum} \underset{ deg = 4k }{ \underbrace{ (-1)^{k} p_{k}(\nabla) } } \\ & \;=\; \underset{k \in \mathbb{N}}{\sum} \underset{ deg = 4k }{ \underbrace{ c_{2k}(\nabla) } } \\ & \;=\; 1 \\ & \phantom{\;=\;} + \underset{ \color{blue} = - p_1(\nabla) }{ \underbrace{ \frac {\tr\big( (F_\nabla)^2 \big) } { 8 \pi^2 } }} \\ & \phantom{\;=\;} + \underset{ \color{blue} = p_2(\nabla) }{ \underbrace{ \frac { - 2 \cdot tr\big( (F_\nabla)^4 \big) + tr\big( (F_\nabla)^2 \big)^2 } {128 \pi^4} }} \\ \phantom{\;=\;} + \cdots \end{aligned}

Hence the first couple of Pontrjagin forms are

p 1() =tr((F ) 2)8π 2 p 2() =tr((F ) 2) 22tr((F ) 4)128π 4. \begin{aligned} p_1(\nabla) & \;=\; - \frac {\tr\big( (F_\nabla)^2 \big) } { 8 \pi^2 } \\ p_2(\nabla) & \;=\; \frac { tr\big( (F_\nabla)^2 \big)^2 - 2 \cdot tr\big( (F_\nabla)^4 \big) } {128 \pi^4} \,. \end{aligned}

(See also, e.g., Nakahara 2003, Exp. 11.5)

Euler forms

For n=2kn = 2k and with the curvature form again regarded as a 2-form valued (2k)×(2k)(2k) \times (2k)-square matrix

F =((F ) a b) 1a,b,2k F_{\nabla} \;=\; \big( (F_{\nabla})^a{}_b \big)_{1 \leq a,b, \leq 2k}

the Euler form is its Pfaffian of this matrix, hence the following sum over permutations σSym(2k)\sigma \in Sym(2k) with summands signed by the the signature sgn(σ){±1}sgn(\sigma) \in \{\pm 1\}:

χ 2k()=(1) k(4π) kk!σsgn(σ)(F ) σ(1)σ(2)(F ) σ(3)σ(4)(F ) σ(2k1)σ(2k). \chi_{2k}(\nabla) \;=\; \frac {(-1)^k} { (4 \pi)^k \cdot k! } \underset{\sigma}{\sum} sgn(\sigma) \cdot (F_{\nabla})_{\sigma(1)\sigma(2)} \wedge (F_{\nabla})_{\sigma(3)\sigma(4)} \wedge \cdots \wedge (F_{\nabla})_{\sigma(2k-1)\sigma(2k)} \,.

The first of these is, using the Einstein summation convention and the Levi-Civita symbol:

χ 4()=ϵ abcd(F ) ab(F ) cd32π 2 \chi_4(\nabla) \;=\; \frac { \epsilon^{ a b c d} (F_{\nabla})_{a b} \wedge (F_\nabla)_{c d} } {32 \pi^2}

(See also, e.g., Nakahara 2003, Exp. 11.7)

Trivializations and structures

The twisted differential c-structures corresponding to Pontryagin class include

References

General

Original accounts:

Classical textbook references are

With an eye towards mathematical physics:

See also

  • Paul Bressler, The first Pontryagin class, math.AT/0509563

  • Ivan Panin, Charles Walter, Quaternionic Grassmannians and Pontryagin classes in algebraic geometry, arxiv/1011.0649

A brief introduction is in chapter 23, section 7

Relation to gravitational instantons

First Pontrjagin class as counting gravitational instanton number:

Last revised on July 25, 2021 at 11:32:14. See the history of this page for a list of all contributions to it.