Euler class

Euler class
Name	Euler class
Field	Algebraic topology
Introduced	20th century
Notable for	Obstruction theory, characteristic classes, index theorems

Euler class

The Euler class is a primary characteristic class associated to oriented real vector bundles, encoding obstruction information for nonvanishing sections and relating topological invariants to analytic and geometric data. It appears in the study of manifolds, fiber bundles, index theorems, and intersection theory, and it connects to historical developments involving figures and institutions such as Hermann Weyl, Marston Morse, Atiyah–Singer index theorem, Élie Cartan, and the Institut des Hautes Études Scientifiques. The class plays a central role in modern treatments of cobordism, fixed point theorems, and moduli problems.

Definition and basic properties

The Euler class is defined for an oriented real vector bundle over a base space and lives in the top-degree cohomology group of the base; for a rank-n bundle over a space X it is an element of H^n(X; Z), where X may be a manifold studied by Henri Poincaré or a CW complex considered by J. H. C. Whitehead. The Euler class is natural with respect to pullbacks along maps studied in the work of Leray and Serre, is functorial in the sense emphasized by Samuel Eilenberg and Saunders Mac Lane, and satisfies a Whitney sum formula relating sums of bundles to cup products appearing in the cohomology theories of Jean Leray and Jean-Pierre Serre. For oriented sphere bundles it coincides with the obstruction class to existence of a nowhere-vanishing section as in obstruction theory developed by P. A. Smith and S. Eilenberg.

Construction for oriented real vector bundles

One standard construction uses the Thom isomorphism for oriented bundles over a base X, building on work of Ralph Fox and René Thom: the Euler class is the image of the Thom class under the restriction map from the relative cohomology of the disk bundle to the cohomology of the base. Alternatively, modelling cohomology via singular cochains influenced by Eilenberg–Steenrod axioms, one constructs the Euler class as the obstruction cocycle for a section extension problem over skeleta of a CW decomposition of X following techniques used by J. H. C. Whitehead and G. W. Whitehead. For smooth bundles over manifolds with structure studied by Elie Cartan and André Weil, the Euler form from Chern–Weil theory yields a de Rham representative of the Euler class; this connects to differential geometers like Shiing-Shen Chern and analysts like Isadore Singer.

Relationship with characteristic classes and cohomology

Within the theory of characteristic classes developed by Raoul Bott and Chern, the Euler class is part of the family that includes Stiefel–Whitney classes and Pontryagin classes; for oriented bundles the reduction mod 2 of the Euler class equals the top Stiefel–Whitney class studied by Eduard Stiefel and Hermann Whitney. The Euler class pairs with the fundamental class of a closed oriented manifold as in Poincaré duality discovered by Henri Poincaré to give the Euler characteristic, a bridge used by Marston Morse in Morse theory and by Michael Atiyah and Isadore Singer in index theory. In generalized cohomology theories such as complex K-theory developed by Atiyah and Bott, there are Euler classes or Euler elements associated to virtual bundles, relating to fixed point formulas of Lefschetz and trace formulas used in the work of Grothendieck.

Computation and examples

For the tangent bundle of a closed oriented surface Σ_g of genus g (studied by Bernhard Riemann and Felix Klein), the Euler class pairs with the fundamental class to give 2-2g, the classical Euler characteristic. For the tautological line bundle over the real projective space RP^n (investigated by E. H. Moore and L. E. J. Brouwer), the Euler class is related to cohomology generators identified in the work of H. Hopf, and vanishing/nonvanishing distinguishes existence of sections via the Borsuk–Ulam theorem linked to Karol Borsuk and Stanislaw Ulam. For sphere bundles associated to oriented rank-n bundles, explicit computations use obstruction cocycles as in classical topology texts by G. W. Whitehead and examples from J. Milnor and J. D. Stasheff. In complex-oriented settings one computes Euler classes via Chern roots and symmetric polynomials used by Weyl and Schur in representation-theoretic contexts.

Applications in topology and geometry

The Euler class detects obstructions to everywhere-nonzero vector fields on spheres, a problem famously connected to the results of Raoul Bott and Adams about vector fields on spheres and the Hopf invariant. It appears in the Gauss–Bonnet theorem proven by Cohn-Vossen and extended by Allendoerfer and Weil and later by Chern to give global formulas relating curvature integrals to topological Euler characteristic. In foliation theory it classifies certain foliated circle bundles studied by Godbillon and Vey, and in rigidity and moduli problems it contributes to invariants used by Mumford, Torelli-type results, and in geometric structures considered by Thurston. The Euler class also plays a role in dynamical fixed point theorems used by Lefschetz and in modern gauge theory contexts developed by Donaldson and Seiberg–Witten.

Generalizations and variations

Generalizations include equivariant Euler classes in the presence of group actions studied by Borel and Atiyah–Bott, Euler classes in generalized cohomology theories like complex cobordism of Novikov and Quillen, and characteristic classes for orbibundles as in the work of Satake and Thurston. There are also relative Euler classes, localized Euler classes used in virtual fundamental class constructions by Li–Tian and Behrend–Fantechi, and secondary invariants related to torsion theories researched by Ray–Singer and Cheeger–Müller. These variations connect the classical Euler class to a broad web of modern developments across topology, geometry, and mathematical physics.

Category:Algebraic topology