nLab Euler class

Idea

The Euler class $\chi$ (or $e$) 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 $X$, evaluated on its fundamental class, is its Euler characteristic $\chi[X]$.

Properties

Cup square

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

(1)$$\chi(E) \smile \chi(E) \;=\; p_k(E) \,.$$

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

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

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:

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

(e.g. Walschap 04, Section 6.4)

Relation to top Chern class

Proposition

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

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

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

For more see at top Chern class.

Poincaré–Hopf theorem

• 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 $X$ be a smooth manifold.

Write

(2)$$\Omega^{2\bullet}(X) \;\; \in \; CAlg_{\mathbb{R}}$$

for the commutative algebra over the real numbers of even-degree differential forms on $X$, 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 $\mathfrak{g}$ be a semisimple Lie algebra (such as $\mathfrak{su}(d)$ or $\mathfrak{so}(d)$) with Lie algebra representation $V \,\in\, Rep_{\mathbb{C}}(\mathfrak{g})$ over the complex numbers of finite dimension $dim_{\mathbb{C}}(V) \,=\, n \,\in\, \mathbb{N}$ (for instance the adjoint representation or the fundamental representation), hence a homomorphism of Lie algebras

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

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

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

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

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

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

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

The multiplicative unit

(4)$$I \;\in\; Mat_{n \times n} \big( \Omega^{2\bullet}(X) \otimes_{\mathbb{R}} \mathbb{C} \big)$$

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

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

(5)$$F_\nabla \,\in\, \Omega^2(X) \otimes_{\mathbb{R}} \mathfrak{g} \xrightarrow{\; \rho \;} \Omega^2(X) \otimes_{\mathbb{R}} End_{\mathbb{C}}(V) \xhookrightarrow{\;\;\;} \Omega^{2\bullet}(X) \otimes_{\mathbb{R}} End_{\mathbb{C}}(V)[t] \;\simeq\; Mat_{n \times n} \Big( \mathbb{C} \otimes_{\mathbb{R}} \Omega^{2}(X) \Big) \,.$$

The formulas

Chern forms

The total Chern form $c(\nabla)$ is the determinant of the sum of the unit (4) with the curvature form (5), and its component in degree $2k$, for $k \in \mathbb{N}$, is the $k$th Chern form $c_k(\nabla)$:

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

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

(6)$$\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 $tr(F_\nabla) = 0$ in these expressions (6) yields the total Pontrjagin form $p(\nabla)$ with degree=$4k$-components the Pontrjagin forms $p_{k}(\nabla)$:

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

Hence the first couple of Pontrjagin forms are

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

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

Euler forms

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

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

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

$$\chi_{2k}(\nabla) \;=\; \frac {(-1)^k} { (4 \pi)^k \cdot k! } \underset{\sigma}{\sum} sgn(\sigma) \cdot (F_{\nabla})_{\sigma(1)\sigma(2)} \wedge (F_{\nabla})_{\sigma(3)\sigma(4)} \wedge \cdots \wedge (F_{\nabla})_{\sigma(2k-1)\sigma(2k)} \,.$$

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

$$\chi_4(\nabla) \;=\; \frac { \epsilon^{ a b c d} (F_{\nabla})_{a b} \wedge (F_\nabla)_{c d} } {32 \pi^2}$$

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

On unit sphere bundles

Proposition

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

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

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

$$\big( S(\pi) \big)^\ast \chi(\nabla_E) \;=\; d \Omega$$

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

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

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

Fiber integration

Proposition

Let

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

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 $S^4 \simeq Spin(5)/Spin(4)$ (this Prop.).

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

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

for instance

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

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

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

(Bott-Cattaneo 98, Lemma 2.1)

Related concepts

• Pontryagin class, Stiefel-Whitney class, Wu class

• one-loop anomaly polynomial

References

General

• Raoul Bott, Loring Tu, Chapter 11 of Differential Forms in Algebraic Topology, Graduate Texts in Mathematics 82, Springer 1982 (doi:10.1007/BFb0063500)

• Allen Hatcher, Euler and Pontryagin classes, section 3.2 in Vector bundles and K-theory (pdf)

• Anant R. Shastri, section 2.1 of Vector bundles and Characteristic Classes (pdf)

• Michael Hutchings, section 5 of Cup product and intersections (pdf)

• Gerard Walschap, chapter 6.3 of: Metric Structures in Differential Geometry, Graduate Texts in Mathematics, Springer 2004 (doi:10.1007/978-0-387-21826-7)

Discussion of fiber integration:

• Raoul Bott, Alberto Cattaneo, Integral Invariants of 3-Manifolds, J. Diff. Geom., 48 (1998) 91-133 (arXiv:dg-ga/9710001, euclid:jdg/1214460608)

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:

• Shiing-Shen Chern, A Simple Intrinsic Proof of the Gauss-Bonnet Formula for Closed Riemannian Manifolds, Annals of Mathematics, Second Series, Vol. 45, No. 4 (1944), pp. 747-752 (jstor:1969302)

• Raoul Bott, Loring Tu, Chapter 11 of Differential Forms in Algebraic Topology, Graduate Texts in Mathematics 82, Springer 1982 (doi:10.1007/BFb0063500)

• Varghese Mathai, Daniel Quillen, below (7.3) of Superconnections, Thom classes, and equivariant differential forms, Topology Volume 25, Issue 1, 1986 (10.1016/0040-9383(86)90007-8)

• Mikio Nakahara, Section 11.4.2 of: Geometry, Topology and Physics, IOP 2003 (doi:10.1201/9781315275826, pdf)

• Walschap 04, section 6.3

• Siye Wu, Section 2.2 of Mathai-Quillen Formalism, pages 390-399 in Encyclopedia of Mathematical Physics 2006 (arXiv:hep-th/0505003)

• Walter A. Poor, 3.58 of Differential Geometric Structures, Dover Books on Mathematics, 2007

• Zhaohu Nie, Secondary Chern-Euler forms and the Law of Vector Fields (arXiv:0909.4754

• Hiro Lee Tanaka, Pfaffians and the Euler class, 2014 (pdf)

• Liviu Nicolaescu, Section 8.3.2 of Lectures on the Geometry of Manifolds, 2018 (pdf, MO comment)