nLab Adams conjecture

Idea

The Adams conjecture is a statement about triviality of spherical fibrations associated to certain formal differences of vector bundles (K-theory classes) via the J-homomorphism. The conjecture was stated in (Adams 63, conjecture 1.2), for vector bundles of rank up to two over a finite CW-complex, which was proven in (Adams 63, theorem 1.4). A general proof was then given in (Quillen 71).

The Adams conjecture/Adams-Quillen theorem serves a central role in the identification of the image of the J-homomorphism.

Statements

Let $X$ be (the homotopy type of) a topological space. For $V \;\colon\; X \longrightarrow B O$ classifying a real vector bundle on $X$, the corresponding spherical fibration is classified by the composite

$$J(V) \;\colon\; X \stackrel{V}{\longrightarrow} B O \stackrel{J}{\longrightarrow} B GL_1(\mathbb{S})$$

with the delooped J-homomorphism. This descends to a map from topological K-theory to spherical fibrations.

Now for $L$ a line bundle on some $X$ and for non-vanishing $k \in \mathbb{Z}$, John Adams observed that the spherical fibration associated with the difference $L^{\otimes k} - L \in K O(X)$ has the property that some $k$-fold multiple of it has trivial spherical fibration, hence that there is $N \in \mathbb{N}$ for which

$$J\left( \oplus^{k^N} (L^{\otimes k} - L) \right) = 0 \,.$$

Noticing that $L \mapsto L^{\otimes^k} = \Psi^k(L)$ is the $k$th Adams operation on K-theory applied to the line bundle $L$, John Adams then conjectured that the above is true for all vector bundles $V$ in the form

$$J\left( \oplus^{k^N} (\Psi^k(V) - V) \right) = 0 \,.$$

Related entries

• J-homomorphism

• Schur index

References

General

The conjecture originates in:

• John Adams, On the groups $J(X)$ I: Topology, 2 (1963) pp. 181–195

Textbook accounts:

• Dai Tamaki, Akira Kono, Section 4.7 in: Generalized Cohomology, Translations of Mathematical Monographs, American Mathematical Society, 2006 (ISBN: 978-0-8218-3514-2)

Review:

• Akhil Mathew, The Adams conjecture I (web)

• Michael Hopkins (notes by Akhil Mathew), Lecture 30 in: Spectra and stable homotopy theory, 2012 (pdf, pdf)

The proof of the Adams conjecture is originally due to

• Daniel Quillen, The Adams conjecture, Topology, vol 10, 1971 (pdf)

The proof using algebraic geometry is due to

• Dennis Sullivan, Genetics of homotopy theory and the Adams conjecture, The Annals of Mathematics, Second Series, Vol. 100, No. 1 (Jul., 1974), pp. 1-79 (JSTOR, pdf)

Yet another proof via Becker-Gottlieb transfer is due to

• J. Becker, D. Gottlieb, The transfer map and fiber bundles Topology , 14 (1975)

In equivariant cohomology

The generalization to equivariant cohomology (equivariant K-theory) is discussed in

• Tammo tom Dieck, theorem 11.3.8 in Transformation Groups and Representation Theory Lecture Notes in Mathematics 766 Springer 1979

• Zbigniew Fiedorowicz, Henning Hauschild, Peter May, Equivariant algebraic K-theory, in: Algebraic K-Theory, Lecture Notes in Mathematics 967, Springer (1982) 23-80 [doi:10.1007/BFb0061898, pdf]

• Henning Hauschild, Stefan Waner, The equivariant Dold theorem mod $k$ and the Adams conjecture, Illinois J. Math. 27 1 (1983) 52-66. [euclid:1256065410&rbrack

• Kuzuhisa Shimakawa, Note on the equivariant $K$-theory spectrum, Publ. RIMS, Kyoto Univ. 29 (1993), 449-453 (pdf, doi)

• Christopher French, theorem 2.4 in The equivariant $J$–homomorphism for finite groups at certain primes, Algebr. Geom. Topol. Volume 9, Number 4 (2009), 1885-1949 (euclid:1513797069)