Euler class

In algebraic topology, the Euler class is a fundamental characteristic class associated with oriented real vector bundles, which assigns to an oriented rank-n vector bundle \xi over a topological space X a cohomology class e(\xi) \in H^n(X; \mathbb{Z}).^[1] This class captures the topological obstruction to the existence of a nowhere-vanishing section of the bundle, generalizing the classical Euler characteristic of manifolds via the pairing e(TX) \cap [X] = \chi(X) for the tangent bundle TX of a closed oriented manifold X.^[2] Defined through the transgression of a Thom class in the cohomology of the sphere bundle or via differential forms on the base space, the Euler class is natural under bundle pullbacks and multiplicative under the Whitney sum operation: e(\xi \oplus \eta) = e(\xi) \cup e(\eta).^[1] It vanishes if the bundle admits a nowhere-vanishing global section and relates to other characteristic classes, such as equaling the top Chern class c_n(V) of a complex vector bundle V, for the underlying real oriented bundle of V, and connecting to Pontryagin classes via e(E)^2 = p_k(E) for even-rank bundles.^[2] Notable examples include twice the generator of H^2(S^2; \mathbb{Z}) for the tangent bundle of the 2-sphere and the role in the Gysin sequence for sphere bundles, highlighting its centrality in computing cohomology rings and invariants.

Definition

Formal Definition

The Euler class of an oriented real vector bundle \xi of rank k over a topological space X is defined as the primary obstruction class to the existence of a nowhere-vanishing section of \xi, residing in the cohomology group H^k(X; \mathbb{Z}).^[3]^[4] More explicitly, let U \in H^k(total space of \xi, complement of zero section; \mathbb{Z}) be the Thom class, which exists due to the orientability of \xi. The Euler class e(\xi) \in H^k(X; \mathbb{Z}) is then the pullback i^*U, where i: X \to total space of \xi is the zero section.^[4]^[5] Note that if the rank k is odd, then e(\xi) is 2-torsion. This definition requires \xi to be oriented to ensure the integral cohomology class is well-defined; for non-oriented bundles, a mod-2 Euler class can be defined using \mathbb{Z}/2\mathbb{Z}-coefficients.^[4][](https://pi.math.cornell.edu/~dmehrle/notes/cornell/17fa/ 6530notes.pdf) In the context of a connection on \xi, if the rank k = 2m is even, the Euler class admits a representative in de Rham cohomology given by e(\xi) = \left[\frac{1}{(2\pi)^m} \mathrm{Pf}(\Omega)\right], where \Omega is the curvature 2-form of the connection and \mathrm{Pf} denotes the Pfaffian. For odd rank, the de Rham representative is 0.^[6]^[7]

Construction Methods

One practical method to construct the Euler class of an oriented real vector bundle \xi of rank k over the k-sphere S^k utilizes the clutching function associated to the bundle. The sphere S^k is decomposed into two hemispheres D^k_+ and D^k_- glued along their equatorial boundary S^{k-1}, and the bundle \xi is trivialized over each hemisphere. The clutching function is then a map f: S^{k-1} \to SO(k) that identifies the fibers over the equator, determining \xi up to isomorphism. The Euler class e(\xi) \in H^k(S^k; \mathbb{Z}) \cong \mathbb{Z} is precisely the degree of this clutching map f, viewed as an element of \pi_{k-1}(SO(k)) under the identification [S^{k-1}, SO(k)]_* \cong \mathbb{Z} for the generator of the homotopy group.^[8] For oriented real vector bundles of rank k, the Euler class can also be realized via the classifying space BSO(k), the base of the universal oriented bundle \gamma^k \to BSO(k). Any such bundle \xi \to B is classified by a map p: B \to BSO(k), and the Euler class is the pullback e(\xi) = p^* e(\gamma^k), where e(\gamma^k) \in H^k(BSO(k); \mathbb{Z}) is the universal Euler class generating this cohomology group via the fundamental class of the Grassmannian. This construction extends the clutching approach universally, as homotopy classes [B, BSO(k)] correspond to isomorphism classes of bundles, with the Euler class capturing the k-th obstruction.^[9]^[10] The Euler class admits an axiomatic characterization as the unique natural transformation from the functor of oriented real vector bundles to cohomology with \mathbb{Z} coefficients that satisfies two properties: the Whitney sum formula e(\xi \oplus \eta) = e(\xi) \cup e(\eta) for bundles over the same base, and a normalization axiom on spheres, where for even k, e(\xi) = \pm 1 \in H^k(S^k; \mathbb{Z}) for the generator corresponding to the Hopf bundle (for k=2) or tautological oriented bundle. For odd k, the Euler class is 2-torsion. This uniqueness ensures that constructions via clutching or classifying spaces yield the same class, providing a foundational tool for computations.^[9]^[11] For a complex vector bundle \xi of rank n, the Euler class of the underlying oriented real bundle of rank $2n is the top Chern class c_n(\xi). In particular, for n=1, e(\xi_\mathbb{R}) = c_1(\xi), the first Chern class.^[12]

Properties

Vanishing Locus of Generic Sections

For an oriented real vector bundle \xi of rank k over a compact oriented manifold M of dimension m \geq k, a generic smooth section s: M \to \xi intersects the zero section transversally, and its zero locus Z(s) = \{x \in M \mid s(x) = 0\} is a smooth closed submanifold of M of dimension m - k.^[9] The orientation on \xi induces an orientation on the normal bundle of Z(s) in M, and the Poincaré dual of [Z(s)] \in H_{m-k}(M; \mathbb{Z}) is e(\xi) \in H^k(M; \mathbb{Z}), where e(\xi) is the Euler class of \xi.^[9] When m = k, the zero locus consists of isolated points, and the evaluation \langle e(\xi), [M] \rangle equals the signed count of these zeros, with signs determined by the local degree of s at each zero (positive if orientation-preserving, negative otherwise).^[9]^[13] This result specializes to the Poincaré–Hopf theorem when \xi = TM is the tangent bundle of an oriented compact manifold M: for a generic vector field (i.e., section of TM) with isolated zeros, the signed count of zeros equals the Euler characteristic \chi(M) = \langle e(TM), [M] \rangle.^[9] The theorem, originally proved by Poincaré in 1885 and Hopf in 1927, thus identifies the Euler class of the tangent bundle with the topological invariant \chi(M).^[9] A necessary condition for the existence of a nowhere-zero section of \xi is that e(\xi) = 0 in H^k(M; \mathbb{Z}), since otherwise the signed count of zeros for any generic section would be nonzero.^[9]^[13] For rank-k bundles over a k-dimensional manifold, this condition is also sufficient: the Euler class is the primary (and only) obstruction in the relevant Postnikov tower for lifting sections from the (k-1)-skeleton to the full space.^[9] In higher-rank cases (e.g., rank k > \dim M), vanishing of e(\xi) is necessary but insufficient, as obstruction theory reveals secondary obstructions in higher cohomology groups; for instance, over a (k+1)-manifold with k \geq 4, a rank-k bundle admits a nowhere-zero section if and only if e(\xi) = 0 and a secondary obstruction o_2(\xi) \in H^{k+1}(M; \pi_k(SO(k))) vanishes.^[14] The transversality of generic sections to the zero section follows from Thom transversality theory: the space of smooth sections is dense in the C^\infty-topology, and perturbations can achieve transversality without altering the homotopy class, ensuring the zero locus represents e(\xi) homologically.^[9] This geometric realization underscores the Euler class as the Poincaré dual to the zero cycle of a generic section.^[9]

Self-Intersection Interpretations

For an oriented real vector bundle \xi of rank k over a smooth manifold X, the projectivization P(\xi) is the bundle whose fiber over each point of X is the real projective space of lines in the fiber of \xi. This construction equips P(\xi) with a tautological real line bundle O(-1) \to P(\xi), whose fiber over a line \ell \subset \xi_x is \ell itself. The Euler class e(\xi) \in H^k(X; \mathbb{Z}) admits a projective geometric interpretation as the self-intersection class of the zero section when viewing the total space of \xi compactified via the projectivization P(\xi \oplus \mathcal{O}_X), where \mathcal{O}_X is the trivial line bundle over X. In this compactification, the zero section j: X \hookrightarrow P(\xi \oplus \mathcal{O}_X) embeds X as a submanifold, and its normal bundle is isomorphic to \xi, yielding the self-intersection class j^* e(N_j) = e(\xi).^[15] To relate this to the geometry of P(\xi), consider the relative Euler sequence for the projectivization:

$0 \to O(-1) \to p^* \xi \otimes O(1) \to T_{P(\xi)/X} \to 0,

where p: P(\xi) \to X is the projection and O(1) is the dual of the tautological line bundle. The Euler class of the relative tangent bundle T_{P(\xi)/X} restricts along the appropriate section (corresponding to the zero direction in the compactified setting) to e(\xi), providing a cohomological link between the twisting of \xi and the geometry of its projectivization. This sequence highlights how the self-intersection arises from the topology of the fibers.^[16] For complex vector bundles, the interpretation simplifies since the Euler class coincides with the top Chern class: if \xi is a complex rank-k bundle, then e(\xi) = c_k(\xi). In the real oriented case, the Euler class is obtained via complexification: e(\xi) = c_k(\xi \otimes \mathbb{C}), where \xi \otimes \mathbb{C} is the complex bundle underlying \xi viewed as a complex vector space of dimension k. This equivalence preserves the self-intersection interpretation, as the top Chern class of the complexification captures the orientable twisting measured by e(\xi).^[12] The pairing \langle e(\xi), [X] \rangle gives the self-intersection index of the zero section, which equals the Euler characteristic of X when \xi = TX is the tangent bundle. This index counts the signed number of intersection points of the zero section with a generic perturbation, reflecting obstructions to sections of \xi.^[16]

Relations to Other Invariants

Connection to Pontryagin Classes

For an oriented real vector bundle \xi of rank $2n over a topological space X, the square of the Euler class equals the nth Pontryagin class: e(\xi)^2 = p_n(\xi) in the cohomology group H^{2n}(X; \mathbb{Z}). This relation holds integrally under the orientability assumption and applies specifically in the top degree for bundles of even rank. A proof proceeds via complexification. The complexified bundle \xi \otimes_{\mathbb{R}} \mathbb{C} decomposes as \xi \otimes_{\mathbb{R}} \mathbb{C} \cong E \oplus \overline{E}, where E is the subbundle of vectors fixed by multiplication by +i and \overline{E} by -i, each of complex rank n. The Pontryagin classes satisfy p_i(\xi) = (-1)^i c_{2i}(\xi \otimes_{\mathbb{R}} \mathbb{C}), so p_n(\xi) = (-1)^n c_{2n}(\xi \otimes_{\mathbb{R}} \mathbb{C}). The top Chern class factors as c_{2n}(\xi \otimes_{\mathbb{R}} \mathbb{C}) = c_n(E) \cdot c_n(\overline{E}), and since c_k(\overline{E}) = (-1)^k c_k(E), this yields c_{2n}(\xi \otimes_{\mathbb{R}} \mathbb{C}) = (-1)^n c_n(E)^2. Thus, p_n(\xi) = c_n(E)^2. For oriented \xi, the Euler class identifies with e(\xi) = c_n(E), confirming e(\xi)^2 = p_n(\xi). This quadratic relation links real-oriented characteristic classes to their complex counterparts and holds in the standard normalization of these invariants. It was discovered by Shiing-Shen Chern in the 1940s as part of his foundational work establishing connections between real and complex characteristic classes.

Instability and Orientability Conditions

The Euler class serves as an unstable characteristic class for vector bundles, meaning its value depends on the specific rank and does not remain invariant under stabilization by adding trivial line bundles. In the stable range, where the bundle rank exceeds the dimension of the base space by a sufficient margin, vector bundles are classified up to stable isomorphism by elements in real or complex K-theory (KO(X) or K(X)), but the Euler class provides an independent cohomological obstruction that may prevent alignment with these stable invariants. A concrete illustration arises in the Atiyah-Hirzebruch spectral sequence converging to KO^*(X), where the Euler class can contribute to non-trivial differentials or extension problems, obstructing the stable triviality of an oriented bundle even when its image in the E_2 page suggests otherwise.^[17]^[18]^[19] The integral Euler class is defined exclusively for oriented real vector bundles, those admitting a consistent choice of orientation across fibers, equivalent to the first Stiefel-Whitney class vanishing: w_1(ξ) = 0. Without orientability, the Euler class cannot be defined over the integers, though a mod 2 analogue exists via reduction to Z/2 coefficients. This mod 2 reduction aligns with the general relation for oriented bundles of rank k, where e(ξ) mod 2 equals the top Stiefel-Whitney class w_k(ξ).^[20] The Euler class vanishes under certain dimensional and topological conditions. Specifically, if the rank of the bundle exceeds the dimension of the base space X, then e(ξ) lies in a cohomology group H^{rk(ξ)}(X; Z) of degree higher than dim(X), hence e(ξ) = 0. Additionally, over contractible base spaces, every vector bundle is trivial, implying e(ξ) = 0, as the Euler class of a trivial bundle is zero. These vanishing theorems underscore the Euler class's role as a primary obstruction in low-dimensional settings, contrasting with stable phenomena where other invariants dominate.^[20]

Examples and Applications

On Spheres

Spheres serve as fundamental examples for computing the Euler class of oriented real vector bundles, providing canonical normalizations due to their simple cohomology rings. For the trivial rank k bundle \varepsilon^k over S^m, the Euler class vanishes, e(\varepsilon^k) = 0, since the bundle admits k linearly independent global sections, implying the existence of a nowhere-zero section.^[13] This holds for any m, as the triviality ensures no topological obstruction to sectioning. In contrast, non-trivial bundles over spheres can have non-zero Euler classes when the rank equals the dimension of the sphere, residing in H^k(S^k; \mathbb{Z}) \cong \mathbb{Z}, generated by the fundamental class [S^k].^[9] A special case arises for the circle S^1, where oriented rank-1 real bundles are necessarily trivial, yielding e = 0. The unique non-trivial rank-1 bundle over S^1 is the Möbius band, which is non-orientable, so its Euler class is undefined over \mathbb{Z}; however, its mod-2 reduction satisfies e \mod 2 = w_1 \neq 0, where w_1 is the first Stiefel-Whitney class, the non-zero generator of H^1(S^1; \mathbb{Z}/2\mathbb{Z}).^[13] For higher even-dimensional spheres, the tangent bundle TS^{2l} exemplifies a non-vanishing Euler class: e(TS^{2l}) = 2 [S^{2l}], twice the generator, reflecting the Euler characteristic \chi(S^{2l}) = 2. For odd-dimensional spheres S^{2l+1}, e(TS^{2l+1}) = 0, consistent with parallelizability and \chi(S^{2l+1}) = 0.^[9] The Hopf bundle over S^2, viewed as the underlying oriented rank-2 real bundle of the tautological complex line bundle on \mathbb{CP}^1 \cong S^2, has Euler class e(\eta) equal to the generator of H^2(S^2; \mathbb{Z}) \cong \mathbb{Z}, so e(\eta) = [S^2] (up to sign convention).^[9] This bundle is constructed via the clutching function of degree 1 on the equator S^1 \to SO(2) \cong S^1. More generally, oriented rank-k bundles over S^k (k \geq 2) are classified by the clutching construction, where a map f: S^{k-1} \to SO(k) of degree d \in \mathbb{Z} (from \pi_{k-1}(SO(k)) \cong \mathbb{Z} for k > 2) yields e(\xi) = d [S^k]. The trivial bundle corresponds to d=0, while the generator case (d=1) provides the normalization for the cohomology ring.^[13]

On Complex Projective Spaces

The Euler class provides a key illustration in the cohomology of complex projective spaces \mathbb{CP}^n, particularly for the tangent bundle T\mathbb{CP}^n, which is an oriented real vector bundle of rank $2n arising from a complex structure. For a complex vector bundle E of rank n, the Euler class e(E) in H^{2n}(B; \mathbb{Z}) coincides with the top Chern class c_n(E), reflecting the oriented nature of the underlying real bundle. The cohomology ring H^*(\mathbb{CP}^n; \mathbb{Z}) is generated by the class h \in H^2(\mathbb{CP}^n; \mathbb{Z}), the positive generator corresponding to the hyperplane line bundle \mathcal{O}(1). The total Chern class of the tangent bundle is c(T\mathbb{CP}^n) = (1 + h)^{n+1}, derived from the Euler sequence $0 \to \mathcal{O}_{\mathbb{CP}^n} \to \mathcal{O}_{\mathbb{CP}^n}(1)^{\oplus (n+1)} \to T\mathbb{CP}^n \to 0 via the Whitney product formula, as the Chern class of \mathcal{O}(1) is $1 + h and that of the trivial bundle is 1. Expanding the binomial yields the top Chern class c_n(T\mathbb{CP}^n) = \binom{n+1}{n} h^n = (n+1) h^n, so e(T\mathbb{CP}^n) = (n+1) h^n. This class is nonzero, indicating that \mathbb{CP}^n admits no nowhere-vanishing section of T\mathbb{CP}^n, consistent with its non-parallelizability for n > 0. Evaluating the Euler class on the fundamental class [\mathbb{CP}^n] gives \langle e(T\mathbb{CP}^n), [\mathbb{CP}^n] \rangle = n+1, which equals the topological Euler characteristic \chi(\mathbb{CP}^n), computed via the cell decomposition of \mathbb{CP}^n into n+1 even-dimensional cells. This pairing exemplifies the general theorem that for a closed oriented $2n-manifold M, \chi(M) = \langle e(TM), [M] \rangle. A related example is the tautological complex line bundle \gamma \to \mathbb{CP}^n, whose dual \gamma^\vee \cong \mathcal{O}(-1) has first Chern class c_1(\gamma^\vee) = -h. As a complex rank-1 bundle, its Euler class is e(\gamma) = c_1(\gamma) = -h \in H^2(\mathbb{CP}^n; \mathbb{Z}), the obstruction to a global nonzero section, which vanishes precisely along the zero section in the total space. The quotient bundle Q = \mathbb{C}^{n+1}/\gamma over \mathbb{CP}^n has Euler class determined by its Chern classes, and the tangent bundle relates via T\mathbb{CP}^n \cong \gamma^\vee \otimes Q, yielding the aforementioned formula upon computing the Chern classes of the tensor product.