Atiyah–Hirzebruch spectral sequence

Atiyah–Hirzebruch spectral sequence
Name	Atiyah–Hirzebruch spectral sequence
Type	Spectral sequence
Introduced	1961
Creators	Michael Atiyah; Friedrich Hirzebruch
Field	Algebraic topology; Topology; K-theory

Atiyah–Hirzebruch spectral sequence The Atiyah–Hirzebruch spectral sequence is a computational tool introduced by Michael Atiyah and Friedrich Hirzebruch linking generalized cohomology theories to ordinary cohomology, used to compute invariants such as topological K-theory and complex cobordism of CW complexes. It organizes information from the skeletal filtration of a CW complex into successive pages, allowing comparisons between Poincaré-style invariants, Cartan operations, and higher differentials discovered in work related to the Adams spectral sequence and Serre spectral sequence.

Definition and statement

The spectral sequence arises for a CW complex X and a generalized cohomology theory E^*, producing an first quadrant spectral sequence with E_2-term expressed in terms of ordinary cohomology groups H^*(X; E^*(pt)). Its formal statement parallels constructions by Jean Leray and Jean-Pierre Serre for fibrations, and it satisfies naturality properties under maps considered by Weyl-style functoriality and Hardy-type exactness. The E_2-term is E_2^{p,q} ≅ H^p(X; E^q(pt)), and differentials d_r have degree (r, 1−r) as in classical treatments by Eilenberg and Mac Lane; convergence is toward graded pieces associated to a filtration on E^{p+q}(X), refining earlier ideas of Cartan and Weil.

Construction and filtration

Construction uses the skeletal filtration X_0 ⊂ X_1 ⊂ ... of a CW complex X, a perspective rooted in the work of Whitehead and developed alongside methods of Alexander duality and Philip S. Hirschhorn-style model categories. One considers the cofiber sequences X_{n−1} → X_n → ∨ S^n indexed by attaching cells analogous to techniques of Thom and Milnor, then applies the generalized cohomology theory E^* using excision principles reminiscent of Lefschetz duality. The resulting exact couples, conceptually dating to Leray and formalized via algebraic frameworks used by Whitehead and Quillen, produce the spectral sequence with an induced filtration F^p E^*(X) = ker(E^*(X) → E^*(X_{p−1})), compatible with multiplicative structures studied by Grothendieck and Cartan.

Convergence and conditionality

Convergence is often conditional: for finite CW complexes the spectral sequence converges strongly to the associated graded of E^*(X), a fact demonstrated in texts influenced by Atiyah and Hirzebruch and further explored in the literature of Adams and Ravenel. For infinite CW complexes one must address lim^1 obstructions related to inverse limit issues analyzed by Serre and May, and the role of connective covers and complete convergence interacts with completions studied by James and Bredon. Conditional convergence phenomena parallel those in the Adams spectral sequence used by Adams to compute stable homotopy groups of spheres studied by Frank Adams and Heinz Hopf.

Calculations and examples

Classical calculations include the computation of K^*(CP^n) using the cellular filtration on complex projective space, a computation linked historically to Hirzebruch–Riemann–Roch and the work of Grothendieck on characteristic classes. The AHSS yields the K-theory of real and complex projective spaces, spheres, and lens spaces studied by Milnor, Stasheff, and Bott; computations for complex cobordism MU^*(X) connect to Thom classes and Stiefel–Whitney and Chern invariants developed by Shiing-Shen Chern. Examples demonstrating nontrivial differentials often reference phenomena discovered by Mahowald and Ravenel, and computations in equivariant contexts use inputs from Graeme Segal and Segal-style equivariant K-theory treated by Atiyah.

Relation to other spectral sequences

The Atiyah–Hirzebruch spectral sequence relates to the Serre spectral sequence for fibrations, to the Adams spectral sequence for stable homotopy, and to the Bockstein spectral sequence arising from exact couples in cohomology with coefficient sequences studied by Eilenberg and Mac Lane. It interacts with the Adams–Novikov spectral sequence used by Sergei Novikov and Ravenel in chromatic homotopy theory, and comparisons to the Lyndon–Hochschild–Serre spectral sequence appear in group cohomology contexts developed by Loday and Cartan. Functoriality and multiplicative structures mirror constructions in the Grothendieck spectral sequence and the Cartan–Eilenberg spectral sequence introduced by Eilenberg.

Applications in topology and K-theory

Applications include computations of topological K-theory groups of manifolds, vector bundles classification influenced by Bott and Atiyah, and computations in index theory related to the Atiyah–Singer index theorem and work by Singer. It is instrumental in detecting torsion classes in generalized cohomology theories relevant to stable homotopy theory and chromatic phenomena studied by Mark Hopkins and Lurie. The AHSS also informs computations in equivariant K-theory for group actions studied by Segal and in orientability questions linked to Edward Witten-inspired elliptic cohomology and TMF studied by Miller and Goerss.

Category:Spectral sequences