Adams spectral sequence

Adams spectral sequence The Adams spectral sequence is a fundamental computational tool in algebraic topology used to compute stable homotopy groups of spheres and other spectra. It relates cohomological Ext groups over the Steenrod algebra to homotopy groups, connecting computational algebraic methods with geometric and categorical structures developed by many contributors in the 20th century. Applications span interactions with homotopy theory, K-theory, cobordism theories, and modern chromatic approaches anchored in work on Morava K-theory and Morava stabilizer group.

Introduction

The Adams spectral sequence arises in the context of computing homotopy groups of spheres via resolutions built from generalized homology and cohomology theories. It is constructed from an Adams resolution using injective or projective models in categories of spectra and relates to derived functors such as Ext in the category of modules over the mod p Steenrod algebra or over MU_*MU bialgebroids. The classical Adams spectral sequence (mod p) uses cohomology with coefficients in Z/pZ and the Steenrod algebra, while the Adams–Novikov spectral sequence replaces ordinary cohomology with complex cobordism MU and the Hopf algebroid (MU_*, MU_*MU).

Construction and algebraic setup

One begins with a ring spectrum or cohomology theory such as H F_p, H Z, MU, BP and forms an Adams resolution of a given spectrum X by choosing maps from wedges of suspensions of the chosen spectrum. The E_2-term is expressed as Ext groups in categories of comodules or modules: for the classical Adams spectral sequence E_2^{s,t} ≅ Ext^{s,t}_{\mathcal A}(H^*(X;Z/p), Z/p) where \mathcal A is the Steenrod algebra. In the Adams–Novikov setup, E_2 ≅ Ext^{s,t}_{MU_*MU}(MU_*(X), MU_*). The algebraic machinery involves computing Ext groups using spectral sequence methods such as the May spectral sequence, bar constructions, and Hopf algebroid cohomology with connections to Koszul duality, Milnor primitives, and Brown–Peterson cohomology structures. Tools from homological algebra like projective resolutions, derived functors, and change-of-rings spectral sequences interplay with computational inputs from the Lambda algebra and techniques introduced by authors working on the cobar complex and Adams resolution theory.

Convergence and applications

Convergence properties depend on completeness and connectivity conditions informed by results from Boardman', Bousfield localization, and work on nilpotence and periodicity by Devinatz, Hopkins, and Smith. The Adams spectral sequence converges conditionally to the p-completed homotopy groups of X under hypotheses such as X being connective and of finite type; variants converge to completions associated to the chosen homology theory, e.g., K-theory completions linked to the Atiyah–Hirzebruch spectral sequence context. Applications include calculations of stable homotopy groups of spheres, insights into exotic elements like Hopf invariant one problem solutions, relations to exotic spheres studied by Kervaire and Milnor, and connections to the Chromatic spectral sequence and Ravenel’s conjectures resolved using Nilpotence theorem techniques. The spectral sequence also informs computations in bordism theory, equivariant topology, and structured ring spectra analyses employed in modern work by groups associated with Institute for Advanced Study and national research institutions.

Computations and examples

Classical computations using the Adams spectral sequence produced the first large-scale charts of the stable homotopy groups of spheres, with foundational computations by J. Frank Adams, Mahowald, Toda, Barratt, Milnor, and Novikov. E_2-term charts are often visualized in Adams charts showing Ext groups and differentials; calculations employ the May spectral sequence and computer-assisted programs developed in collaborative projects including work at University of Chicago, Princeton University, and University of Cambridge. Specific examples include computation of the 2-primary and 3-primary components of \pi_*^s, identification of families such as the alpha family, beta family, and detection of periodicities explained by Morava theories and v_n-periodicity. The Adams–Novikov spectral sequence yields more refined inputs using BP or MU cooperations and led to calculations of complex cobordism rings and modular forms appearing in topological modular forms studies pursued at institutions like Harvard University and MIT.

Variants and generalizations

Numerous variants generalize the Adams construction: the Adams–Novikov spectral sequence, the Adams spectral sequence based on generalized homology theories, and equivariant Adams spectral sequences for G-spectra with groups such as C_2 or finite groups. Chromatic versions incorporate Morava E-theory and the chromatic spectral sequence, while localized or completed forms use Bousfield localization and p-completion. Derived and motivic analogues adapt the Adams machinery to contexts studied in motivic homotopy theory, A^1-homotopy theory, and in arithmetic settings connected to algebraic K-theory and etale cohomology. Recent generalizations involve spectral algebraic geometry tools developed in projects at Institute for Advanced Study and collaborations across Mathematical Sciences Research Institute and leading universities.

Historical development and key contributors

The spectral sequence was introduced and developed by J. Frank Adams in the 1950s with foundational studies published during collaborations and correspondence with contemporaries. Significant contributors include Novikov, who introduced cobordism-based adaptations, Milnor and Moore for algebraic structures in cohomology operations, and Margolis for module-theoretic perspectives. Later advances by Ravenel, Hopkins, Smith, Devinatz, Mahowald, Toda, Barratt, Dwyer, Kan, and Bousfield shaped convergence, nilpotence, and localization results. Institutional centers for development included University of Cambridge, Princeton University, University of Chicago, and University of California, Berkeley, with contemporary work continuing in research groups across Harvard University, Massachusetts Institute of Technology, Institut des Hautes Études Scientifiques, and other global centers of algebraic topology. Category:Algebraic topology