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Euler class

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Context

Algebraic topology

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

The Euler class χ\chi (or ee) is a characteristic class of the special orthogonal group, hence of oriented real vector bundles.

The Euler class of the tangent bundle of a smooth manifold XX, evaluated on its fundamental class, is its Euler characteristic χ[X]\chi[X].

Properties

Cup square

For EE a vector bundle of even rank rank(E)=2krank(E) = 2 k, the cup product of the Euler class with itself equals the kkth Pontryagin class

(1)χ(E)χ(E)=p k(E). \chi(E) \smile \chi(E) \;=\; p_k(E) \,.

(e.g. Walschap 04, Section 6.3, p. 187)

When the Euler class is represented by the Euler form of a connection \nabla on EE, which then is fiber-wise proportional to the Pfaffian of the curvature form F F_\nabla of \nabla, the relation (1) corresponds to the fact that the product of a Pfaffian with itself is the determinant: (Pf(F )) 2=det(F )\big( Pf(F_\nabla) \big)^2 = det(F_\nabla).

Whitney sum formula

Proposition

(Euler class takes Whitney sum to cup product)

The Euler class of the Whitney sum of two oriented real vector bundles to the cup product of the separate Euler classes:

χ(EF)=χ(E)χ(F). \chi( E \oplus F ) \;=\; \chi(E) \smile \chi(F) \,.

(e.g. Walschap 04, Section 6.4)

Relation to top Chern class

Proposition

The top Chern class of a complex vector bundle 𝒱 X\mathcal{V}_X equals the Euler class ee of the underlying real vector bundle 𝒱 X \mathcal{V}^{\mathbb{R}}_X:

𝒱 Xhas complex ranknc n(𝒱 X)=e(𝒱 X )H 2n(X;). \mathcal{V}_X \; \text{has complex rank}\;n \;\;\;\;\; \Rightarrow \;\;\;\;\; c_n \big( \mathcal{V}_X \big) \;\; = \;\; e \big( \mathcal{V}^{\mathbb{R}}_X \big) \;\;\;\; \in H^{2n} \big( X; \, \mathbb{Z} \big) \,.

(e.g. Bott-Tu 82 (20.10.6))

For more see at top Chern class.

Poincaré–Hopf theorem

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

(2)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

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

The tensor product of the \mathbb{C}-algebras (2) and (3)

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

(4)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

(5)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 (4) with the curvature form (5), 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}:

(6)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 (6) 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)

On unit sphere bundles

Proposition

Let XX be a smooth manifold and EπXE \overset{\pi}{\longrightarrow} X an oriented real vector bundle of even rank, rank(E)=2k+2rank(E) = 2k + 2.

For any choice of connection \nabla on EE (SO(dim(X))SO(dim(X))-connection), let χ( E)Ω 2k(X)\chi(\nabla_E) \in \Omega^{2k}(X) denote the corresponding Euler form.

Then the pullback of the Euler form χ( E)\chi(\nabla_E) to the unit sphere bundle S(E)S(π)XS(E) \overset{S(\pi)}{\longrightarrow} X is exact

(S(π)) *χ( E)=dΩ \big( S(\pi) \big)^\ast \chi(\nabla_E) \;=\; d \Omega

such that the trivializing form has (minus) unit integral over any of the (2k+1)-sphere-fibers S x 2k+1ι xS(E)S^{2k+1}_x \overset{\iota_x}{\hookrightarrow} S(E):

(7) S 2k+1ι x *Ω=1. \int_{S^{2k+1}} \iota_x^\ast \Omega \;=\; -1 \,.

(e.g. Walschap 04, Chapter 6.6, Thm. 6.1, p. 201-202, Poor 07, 3.68, Nie 09)

Fiber integration

Proposition

Let

S 4 BSpin(4) π BSpin(5) \array{ S^4 &\longrightarrow& B Spin(4) \\ && \big\downarrow^{\mathrlap{\pi}} \\ && B Spin(5) }

be the spherical fibration of classifying spaces induced from the canonical inclusion of Spin(4) into Spin(5) and using that the 4-sphere is equivalently the coset space S 4Spin(5)/Spin(4)S^4 \simeq Spin(5)/Spin(4) (this Prop.).

Then the fiber integration of the odd cup powers χ 2k+1\chi^{2k+1} of the Euler class χH 4(BSpin(4),)\chi \in H^4\big( B Spin(4), \mathbb{Z}\big) (see this Prop) are proportional to cup powers of the second Pontryagin class

π *(χ 2k+1)=2(p 2) kH 4(BSpin(5),), \pi_\ast \left( \chi^{2k+1} \right) \;=\; 2 \big( p_2 \big)^k \;\;\in\;\; H^4\big( B Spin(5), \mathbb{Z} \big) \,,

for instance

π *(χ) =2 π *(χ 3) =2p 2 π *(χ 5) =2(p 2) 2H 4(BSpin(5),); \begin{aligned} \pi_\ast \big( \chi \big) & = 2 \\ \pi_\ast \left( \chi^3 \right) & = 2 p_2 \\ \pi_\ast \left( \chi^5 \right) & = 2 (p_2)^2 \end{aligned} \;\;\in\;\; H^4\big( B Spin(5), \mathbb{Z} \big) \,;

while the fiber integration of the even cup powers χ 2k\chi^{2k} vanishes

π *(χ 2k)=0H 4(BSpin(5),). \pi_\ast \left( \chi^{2k} \right) \;=\; 0 \;\;\in\;\; H^4\big( B Spin(5), \mathbb{Z} \big) \,.

(Bott-Cattaneo 98, Lemma 2.1)

References

General

Discussion of fiber integration:

Discussion for projective modules

  • Satya Manda, An overview of Euler class theory (pdf)

See also

  • Wikipedia Euler class

  • Robert F. Brown, On the Lefschetz number and the Euler class, Transactions of the AMS 118, (1965) (JSTOR)

  • Solomon Jekel, A simplicial formula and bound for the Euler class, Israel Journal of Mathematics 66, n. 1-3, 247-259 (1989)

Euler forms

Discussion of Euler forms (differential form-representatives of Euler classes in de Rham cohomology) as Pfaffians of curvature forms:

Last revised on July 26, 2021 at 08:58:12. See the history of this page for a list of all contributions to it.