Adams conjecture

In [a1], J.F. Adams was studying the order of the image of the $J$-homomorphism and related topics. He was led to the following conjecture: Let $F$ be a real vector bundle over a finite CW-complex $X$, let $k$ be an integer and let $\psi$ be the Adams operation (cf. Cohomology operation). Then for some $n$, $k^n(\psi^kF-F)$ is fibre homotopy trivial. In [a4], D.G. Quillen proposed an approach to this conjecture which used some ideas from algebraic geometry and, in particular, étale homotopy theory. This approach was completed by E. Friedlander [a3]. In [a5], Quillen solved the conjecture by a different approach. This approach lead to important results in algebraic $K$-theory and lead to Quillen's definition of higher algebraic $K$-groups. Another proof was discovered by J. Becker and D. Gottlieb [a2]. Their proof uses only techniques from algebraic topology and so could be considered elementary. They introduced the transfer into fibre bundle theory (cf. also Becker–Gottlieb transfer). Previously, transfers required that the fibre be discrete. This work, too, has had a lot of influence. The Adams conjecture has generated a lot of very interesting mathematics.

References

[a1]	J.F. Adams, "On the groups $J(X)$. I" Topology , 2 (1963) pp. 181–195 MR0159336
[a2]	J. Becker, D. Gottlieb, "The transfer map and fiber bundles" Topology , 14 (1975) pp. 1–12 MR0377873 Zbl 0306.55017
[a3]	E. Friedlander, "Fibrations in etale homotopy theory" IHES Publ. Math. , 42 (1972) MR0366929 Zbl 0351.55011
[a4]	D.G. Quillen, "Some remarks on etale homotopy theory and a conjecture of Adams" Topology , 7 (1968) pp. 111–116 MR0227988 Zbl 0157.30303
[a5]	D.G. Quillen, "The Adams conjecture" Topology , 10 (1971) pp. 67–80 MR0279804 Zbl 0219.55013