Adamsnovikov Spectral SequenceEdit
The Adams–Novikov spectral sequence (ANSS) is a foundational computational tool in stable homotopy theory. It refines the classical Adams spectral sequence by replacing ordinary mod p cohomology with a complex-oriented cohomology theory built from formal group laws, most prominently Brown–Peterson theory. The upshot is a powerful framework for organizing and executing calculations of the stable homotopy groups of spheres, especially by exposing chromatic structure that mirrors height phenomena in formal group laws. In practice, the ANSS translates topological problems into the language of algebra: Ext groups over a Hopf algebroid associated with a complex-oriented theory encode the stages of a filtration that converges, under suitable conditions, to the sought-after homotopy groups.
From a results-driven perspective, the ANSS is prized for its ability to reveal deep regularities in the stable homotopy groups and to isolate families of elements according to their height with respect to formal group laws. This makes it a natural home for chromatic homotopy theory, where one studies phenomena at different chromatic levels using Morava K-theories and related constructions. The interplay between topology and algebra here is a defining feature: the E2-term is an Ext group in the category of BP_*BP-comodules, and the subsequent differentials encode intricate higher-order relations among cocycles. The framework also connects to classical objects like the sphere spectrum sphere spectrum and to a network of cohomology theories built from complex cobordism.
Historically, the ANSS sits at the crossroads of two streams. The original Adams spectral sequence, built from mod p cohomology, provided a computational backbone for early stable homotopy calculations. The advent of complex-oriented theories in the 1960s and 1970s—culminating in the theory of formal group laws and the work on BP and complex cobordism—paved the way for Novikov and others to refine the filtration. The result was the Adams–Novikov spectral sequence, which has since become a central instrument in modern stable homotopy theory, alongside the broader project of chromatic homotopy theory chromatic homotopy theory.
Construction and input data
- The input spectrum and theory: The ANSS is built from a complex-oriented cohomology theory, most commonly the Brown–Peterson spectrum Brown-Peterson cohomology and its associated coefficient theory BP* BPBP as a Hopf algebroid. The relevant algebraic object is the Hopf algebroid (BP_*BP, BP_), which encodes the cooperations in BP and the action of its associated formal group law.
- E2-term: The second page of the spectral sequence is an Ext group, E2^{s,t} ≅ Ext^{s,t}{BPBP}(BP_, BP_). This Ext group measures extensions of BP_ comodules and encodes algebraic information that will be reflected in the stable homotopy groups after convergence.
- Convergence and filtration: Under suitable hypotheses (e.g., p-localization, completion, and favorable connectivity), the ANSS converges to the p-local stable homotopy groups of spheres π_*^S. The filtration provided by the spectral sequence often corresponds to chromatic height, connecting to various Morava theories Morava K-theory and to the broader chromatic filtration.
Chromatic perspective and computational payoff
- Height and Morava theories: The ANSS interacts closely with the chromatic filtration, where different heights detect different layers of periodicity in the stable homotopy groups. Morava K-theories provide sharp probes at each height, and the ANSS helps organize contributions from these layers into a coherent computational picture. See Morava K-theory for related viewpoints.
- Practical computations: In practice, calculations in the ANSS often proceed by identifying patterns predicted by the BP_*BP comodule structure, then translating those patterns into potential differentials and extensions in the spectral sequence. At small primes p (for example p = 2 or p = 3), detailed charts illustrate how v1- and v2-periodic phenomena appear and interact, with deep connections to the classical charts coming from the original Adams spectral sequence as well as to later chromatic refinements.
Relationship to broader theory and notable features
- Formal group laws and BP: The BP framework encodes information about formal group laws that govern complex-oriented theories. This algebraic backbone is what enables the Ext computations that feed the E2-term. See formal group law and complex cobordism for foundational background.
- Hopf algebroids and comodule structure: The algebraic side rests on the theory of Hopf algebroids, which describe cooperations in cohomology theories. Studying Ext over BP_*BP reveals the algebraic shadows of topological phenomena. See Hopf algebroid for a concise algebraic account.
- Convergence and extensions: While the ANSS provides a principled route to π_*^S, there are technical subtleties around convergence and hidden extensions. Researchers must carefully distinguish genuine differentials from artifacts of the algebraic setup and account for potential multiplicative and additive extensions in the target groups.
Controversies and debates
- Abstraction vs calculational utility: A common debate centers on the level of abstraction required by the ANSS. Proponents argue that the BP-based, chromatic framework organizes otherwise unwieldy computations and reveals structural regularities in stable homotopy groups. Critics contend that the machinery is highly technical and may obscure concrete computations, especially for newcomers. From a pragmatic, results-oriented angle, supporters emphasize that the clarity it provides about periodicity and height phenomena justifies the complexity, while detractors point to the steep learning curve and the occasional difficulty of turning algebraic data into explicit homotopy classes.
- Convergence caveats and extensions: There are legitimate concerns about when and how the ANSS converges to the full π_*^S and how to interpret hidden extensions. The field has developed a sophisticated vocabulary—including refinements and comparisons with other spectral sequences—to address these issues, but debates persist about the limits of what ANSS computations can certify without supplementary input.
- The role of deep machinery in a broader program: Since the ANSS sits inside chromatic homotopy theory and is tied to large-scale geometric ideas (including the nilpotence and periodicity theorems), some commentators question whether such heavy machinery should be indispensable for meaningful progress in stable homotopy theory. Advocates reply that the depth of the structure uncovered by ANSS is precisely what makes it possible to access chromatic phenomena that simpler approaches cannot illuminate.
- Woke criticisms and defences (in context): In broader mathematical discourse, some observers push back against perceptions of gatekeeping or gatekeeping-style critiques—arguing that the field’s advancement relies on specialized tools and collaborative effort, not on fashion or ideological gatekeeping. From a right-of-center, results-focused stance, the defense is that advancing rigorous, structural understanding of fundamental objects—like π_*^S—has tangible payoff in mathematics, physics, and related areas, and that criticisms framed as dismissals of expertise tend to miss the value of deep, well-supported theory. Proponents emphasize that focusing on concrete calculations, testable predictions, and robust theorems should guide judgment, rather than stylistic objections about who participates in the discipline.
Notable connections and further directions
- Nilpotence and periodicity: The ANSS interacts with the broader program of nilpotence and periodicity in stable homotopy theory, notably through the nilpotence theorem and related chromatic phenomena. See nilpotence theorem for foundational results in this direction and Hopkins–Ravenel–Smith nilpotence theorem for the culminating synthesis.
- Higher chromatic layers: Beyond BP, the spectrum of ideas extends to more exotic theories and their corresponding spectral sequences. Morava stabilizer groups and related structures enter the story as one passes to higher heights, connecting topology with aspects of algebraic geometry. See Morava stabilizer group and chromatic homotopy theory for broader context.
- Interactions with cohomology theories: The ANSS is part of a web that includes classical tools like the original Adams spectral sequence and modern, highly structured approaches to stable homotopy theory. See Adams spectral sequence for historical context and comparison.