Wu classes

Wu classes
Name	Wu classes
Notation	v_i
Field	Topology
Introduced	1950s
Introduced by	Wu Wenjun

Wu classes are characteristic cohomology classes v_i associated to smooth manifolds and to vector bundles defined over them, encoding operations of Steenrod squares and relations among Stiefel–Whitney classes. They appear in the interplay between cohomological operations studied by W. S. Massey, John Milnor, Hassler Whitney, and work initiated by Wu Wenjun; they are central to calculations in algebraic topology involving characteristic classes, cobordism, and embedding problems. Wu classes bridge classical results of Élie Cartan, Jean-Pierre Serre, and Norman Steenrod with later developments by Raoul Bott, Michael Atiyah, Frank Adams, and Daniel Quillen.

Definition and basic properties

For a closed n-dimensional smooth manifold M the Wu class v = 1 + v_1 + v_2 + ... is a total cohomology class in H^*(M; Z/2) characterized by relations with Steenrod squares introduced by Norman Steenrod and formalized in work of Wu Wenjun. Each Wu class v_i ∈ H^i(M; Z/2) satisfies the property that for every α ∈ H^{n−i}(M; Z/2) one has ⟨v_i ∪ α, [M]⟩ = ⟨Sq^i(α), [M]⟩, connecting evaluation on the fundamental class [M] studied in papers by Hassler Whitney and orientations considered by Jean Leray. The total Wu class depends functorially on the cobordism class of M as in results by René Thom and interacts naturally with cup product and pullbacks under maps treated by Raymond L. Wilder and Samuel Eilenberg.

Wu formula and Wu classes for manifolds

The Wu formula expresses Stiefel–Whitney classes w = 1 + w_1 + w_2 + ... of the tangent bundle of M in terms of the Wu classes v and Steenrod operations studied by Norman Steenrod and Edwin H. Spanier. Explicitly, for the total classes one has w = Sq(v), where Sq denotes the total Steenrod square operator formalized by Norman Steenrod and applied in contexts by Jean-Pierre Serre. Equivalently, individual relations w_k = Σ_{i} Sq^{k−i}(v_i) mirror identities used in the work of Raoul Bott on characteristic classes and those in John Milnor and James D. Stasheff on obstruction theory. These relations allow recovery of the Stiefel–Whitney classes from Wu classes for smooth closed manifolds and underpin vanishing statements exploited by Michael Atiyah in his studies of K-theory and by Vladimir Arnold in singularity theory.

Relation to Steenrod operations and Stiefel–Whitney classes

By definition Wu classes encode the failure of conjugation between evaluation and Steenrod operations; they are designed so that Sq^i is adjoint to cup product with v_i under Poincaré duality associated to the fundamental class studied by René Thom. The interplay between Sq operators and Stiefel–Whitney classes appears in foundational texts by Norman Steenrod and later expositions by Milnor and Stasheff; Wu classes supply the missing input to invert these relations in manifold contexts studied by Wu Wenjun himself. In cobordism and immersion problems considered by Ralph L. Cohen and Barry Mazur, Wu classes provide obstruction classes parallel to those given by Pontryagin classes examined by Lev Pontryagin and the Hirzebruch signature theorem developed by Friedrich Hirzebruch.

Computation and examples

Computing Wu classes often reduces to computing Steenrod squares and Stiefel–Whitney classes for explicit manifolds: for real projective spaces considered by Hassler Whitney and Eduard Study, the Wu classes can be read off from the known structure of H^*(RP^n; Z/2) and the action of Sq studied by J. F. Adams. For complex projective spaces examined by Mihály Csíkszentmihályi and René Thom the vanishing of odd Stiefel–Whitney classes yields explicit simplifications; products of spheres treated by Stephen Smale and John Milnor give Wu classes concentrated in top degrees. Lens spaces and homogeneous spaces studied in work by Marston Morse and Élie Cartan provide further tractable examples using the transfer and spectral sequence techniques of Jean Leray and Jean-Pierre Serre. Algorithmic approaches to compute v_i for bundles and manifolds use Wu’s formula together with the Cartan formula for Steenrod operations as developed in computational topology programs influenced by Serge Lang and implementations inspired by work of James Davis.

Generalizations and applications

Wu classes generalize to Poincaré complexes and to oriented cobordism theories treated by René Thom; they admit analogues in complex cobordism studied by Daniel Quillen and in Morava K-theory considered by Jack Morava. In surgery theory and classification of manifolds pursued by William Browder and C. T. C. Wall, Wu classes appear in surgery obstructions and intersection form considerations linked to Frederick J. Almgren and applications to embedding problems investigated by Ralph Fox. In algebraic geometry, analogues of Wu classes occur in étale cohomology contexts explored by Alexander Grothendieck and in motivic cohomology frameworks connected to work of Vladimir Voevodsky and Spencer Bloch. They also play roles in string topology and field-theoretic invariants where researchers such as Graeme Segal and Edward Witten relate characteristic operations to anomaly cancellation and orientation data.

History and development of the concept

Wu classes were introduced by Wu Wenjun in the 1950s building on infrastructure laid by Norman Steenrod on cohomology operations and by Hassler Whitney on characteristic classes. Early applications appeared in the hands of René Thom and John Milnor in cobordism and manifold classification problems; later systematic treatments were given in textbooks and papers by Milnor and Stasheff, Raoul Bott, and Michael Atiyah. The development continued through interactions with surgery theory led by William Browder and characteristic class refinements in K-theory by Friedrich Hirzebruch and Atiyah–Singer index theory initiatives involving Isadore Singer. Contemporary research connects Wu classes to motivic homotopy theory and to computations in chromatic homotopy theory driven by Douglas Ravenel and Haynes Miller.

Category:Characteristic classes