Overview
The Adams spectral sequence is a computational tool in algebraic topology used to determine the stable homotopy groups of spheres and, more generally, to study the homotopy groups of spectra. Introduced by J. Frank Adams in the 1950s, it relates these homotopy groups to Ext groups in the category of comodules over a Hopf algebroid associated with a chosen homology theory, most commonly mod p cohomology.
Construction
Let $E$ be a homology theory represented by a ring spectrum (e.g., ordinary mod p cohomology $H\mathbb{F}_p$ or complex cobordism $MU$). For a spectrum $X$, the $E$-based Adams spectral sequence has
$$ E_2^{s,t} \cong \operatorname{Ext}^{s,t}{\mathcal{A}}(E* X, E_*), $$
where $\mathcal{A}$ denotes the Hopf algebroid $(E_*, E_*E)$ and $\operatorname{Ext}$ is computed in the category of graded comodules over $\mathcal{A}$. The spectral sequence converges, under suitable connectivity hypotheses, to the graded pieces of the $E$-localized homotopy groups of $X$:
$$ E_\infty^{s,t} ;\Longrightarrow; \pi_{t-s}(L_E X). $$
For the classical (mod p) Adams spectral sequence, $\mathcal{A}$ is the Steenrod algebra $\mathcal{A}_p$ and $E = H\mathbb{F}_p$. The $E_2$-term becomes
$$ E_2^{s,t} = \operatorname{Ext}^{s,t}_{\mathcal{A}_p}(\mathbb{F}_p, \mathbb{F}_p), $$
which is a bi‑graded algebra computable via algebraic methods (e.g., the May spectral sequence).
Convergence and Filtration
The spectral sequence is conditionally convergent; it is strongly convergent when $X$ is a finite spectrum or when $E$ satisfies certain nilpotence conditions. The filtration on $\pi_*(X)$ induced by the spectral sequence is called the Adams filtration, measuring the complexity of elements in terms of how many extensions are needed to detect them.
Variants
Several extensions and variants exist:
- Adams–Novikov spectral sequence – uses complex cobordism $MU$ (or Brown–Peterson homology $BP$) as the coefficient theory, yielding a finer filtration and connecting to formal group law theory.
- Motivic Adams spectral sequence – defined in the setting of motivic homotopy theory, employing motivic cohomology as the coefficient theory.
- Chromatic Adams spectral sequences – based on localized homology theories such as $K(n)$ or $E_n$, reflecting the chromatic filtration of stable homotopy.
Applications
The Adams spectral sequence has been central to many breakthroughs:
- Computation of the first few stable homotopy groups of spheres, including the detection of the Hopf invariant one elements and the existence of exotic elements such as the Kervaire invariant classes.
- Proofs of periodicity phenomena (e.g., the existence of $v_n$-periodic families) via the Adams–Novikov spectral sequence.
- Development of the nilpotence and periodicity theorems of Devinatz–Hopkins–Smith, which rely on the behavior of elements in the Adams–Novikov spectral sequence.
- Calculations in the homotopy of structured ring spectra, such as $MU$, $BP$, and various $E_\infty$-ring spectra.
Historical Context
J. F. Adams introduced the spectral sequence in his seminal papers “On the non‑existence of elements of Hopf invariant one” (Ann. of Math., 1958) and “Stable homotopy and generalised homology” (University of Chicago Press, 1974). The method built on earlier work of Atiyah–Hirzebruch on generalized homology theories and on Cartan–Eilenberg’s homological algebra.
References
- J. F. Adams, Stable Homotopy and Generalised Homology, University of Chicago Press, 1974.
- J. F. Adams, “On the non‑existence of elements of Hopf invariant one”, Ann. of Math. 72 (1960), 20–104.
- D. C. Ravenel, Complex Cobordism and Stable Homotopy Groups of Spheres, Academic Press, 1986.
- H. Miller, “The Adams Spectral Sequence and Its Applications”, Proceedings of the International Congress of Mathematicians, 1970.
- M. A. Mandell, J. P. May, S. Schwede, B. Shipley, Model Categories of Diagram Spectra, 2001 (provides modern categorical foundations).
Note: The description above reflects established mathematical literature up to the present date.