View source for Adams conjecture

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In [[#References|[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|vector bundle]] over a finite [[CW-complex|CW-complex]] $X$, let $k$ be an integer and let $\psi$ be the Adams operation (cf. [[Cohomology operation|Cohomology operation]]). Then for some $n$, $k^n(\psi^kF-F)$ is fibre homotopy trivial. In [[#References|[a4]]], D.G. Quillen proposed an approach to this conjecture which used some ideas from [[Algebraic geometry|algebraic geometry]] and, in particular, étale [[Homotopy|homotopy]] theory. This approach was completed by E. Friedlander [[#References|[a3]]]. In [[#References|[a5]]], Quillen solved the conjecture by a different approach. This approach lead to important results in [[Algebraic K-theory|algebraic $K$-theory]] and lead to Quillen's definition of higher algebraic $K$-groups. Another proof was discovered by J. Becker and D. Gottlieb [[#References|[a2]]]. Their proof uses only techniques from [[Algebraic topology|algebraic topology]] and so could be considered elementary. They introduced the transfer into fibre bundle theory (cf. also [[Becker–Gottlieb transfer|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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.F. Adams, "On the groups $J(X)$. I" ''Topology'' , '''2''' (1963) pp. 181–195 {{MR|0159336}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J. Becker, D. Gottlieb, "The transfer map and fiber bundles" ''Topology'' , '''14''' (1975) pp. 1–12 {{MR|0377873}} {{ZBL|0306.55017}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> E. Friedlander, "Fibrations in etale homotopy theory" ''IHES Publ. Math.'' , '''42''' (1972) {{MR|0366929}} {{ZBL|0351.55011}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> D.G. Quillen, "Some remarks on etale homotopy theory and a conjecture of Adams" ''Topology'' , '''7''' (1968) pp. 111–116 {{MR|0227988}} {{ZBL|0157.30303}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> D.G. Quillen, "The Adams conjecture" ''Topology'' , '''10''' (1971) pp. 67–80 {{MR|0279804}} {{ZBL|0219.55013}} </TD></TR></table>