Math 2.0

Algebraic topology Theorem

Adams conjecture

Builds on Adams operationsReal K-theory KOSphere bundleJ-homomorphismCW complexStable homotopy groups of spheres

Statement

Let XX be a finite CW complex. Let KO~(X)\widetilde{KO}(X) be reduced real K-theory. Let J(X)J(X) be its quotient by the subgroup generated by the classes [E]−[F][E] - [F] of the following kind. For some m,nm, n with rank⁡E+m=rank⁡F+n\operatorname{rank} E + m = \operatorname{rank} F + n, the sphere bundles of E⊕ℝmE \oplus \R^m and F⊕ℝnF \oplus \R^n are fibre homotopy equivalent. Let ψk\psi^k be the kk-th Adams operation on KO~(X)\widetilde{KO}(X). Then for every integer kk and every x∈KO~(X)x \in \widetilde{KO}(X) there is an integer e≥0e \ge 0 such that ke(ψk−1)(x)=0in J(X).k^e (\psi^k - 1)(x) = 0 \quad \text{in } J(X).

Remarks

Adams stated this conjecture in his series of papers on the groups J(X)J(X), and proved it in special cases. The full statement was proved by Daniel Quillen and by Dennis Sullivan, and later also by Becker and Gottlieb with the transfer. The proofs of Quillen and of Sullivan use ideas from algebraic geometry over fields of positive characteristic and from étale homotopy theory.

The theorem gives an upper bound for J(X)J(X), and Adams had found a matching lower bound using the e-invariant. Together they determine J(X)J(X) for many XX, in particular for spheres, which gives the image of the stable J-homomorphism in the stable homotopy groups of spheres. For n=4m−1n = 4m - 1 with m≥1m \ge 1, this image is cyclic of order equal to the denominator of B2m/4mB_{2m}/4m, where B2mB_{2m} is a Bernoulli number.

Going deeper

For a student

This is the key step in the computation of the image of the J-homomorphism, which is the most regular and best understood part of the stable homotopy groups of spheres. The denominators of Bernoulli numbers that give its order connect homotopy theory with number theory. The proofs led to new methods, such as the use of étale homotopy and of transfers for fibrations.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

Show the path as a graph
Adams operations, concept, no evidence levelAdams operationsconceptCW complex, concept, no evidence levelCW complexconceptJ-homomorphism, concept, no evidence levelJ-homomorphismconceptReal K-theory KO, concept, no evidence levelReal K-theory KOconceptSphere bundle, concept, no evidence levelSphere bundleconceptStable homotopy groups of spheres, concept, no evidence levelStable homotopy group…conceptAdams conjecture, statement, no evidence levelAdams conjecturestatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Adams operations concept
  2. CW complex concept
  3. J-homomorphism concept
  4. Real K-theory KO concept
  5. Sphere bundle concept
  6. Stable homotopy groups of spheres concept
  7. Adams conjecture statement

Further reading

  1. Web https://en.wikipedia.org/wiki/J-homomorphism
About this page
TagX12WG9TPQ0
URIm2://X12WG9TPQ0
Kindstatement, theorem
WritingW0 stub
MSC55Q50, 19L20
StatusThis copy of the site has no build data. Run m2 status X12WG9TPQ0.
AboutJ-homomorphism, Adams operations
UsesAdams operations, Real K-theory KO, Sphere bundle, J-homomorphism, CW complex, Stable homotopy groups of spheres
Changed3 November 2026

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