University of David
First, the Adams conjecture was almost simultaneously proved by Dennis Sullivan, also using Grothendieck’s theory, but in a different way. While Quillen’s proof led to algebraic K-theory, Sullivan’s was part of a quite different program, his determination of the structure of piecewise linear and topologicalmanifolds. Thiswas just one of several places where Quillen’s work intersected with Sullivan’s though they were proceeding in different directions. Another was their independent development of rational homotopy theory, where Sullivan was motivated by explicit questions about the groups of homotopy equivalences of manifolds. Ib Madsen has remarked on the strange quirk of mathematical history that, a few years later, Becker and Gottlieb found a verymuch more elementary proof of the Adams conjecture which did not use Grothendieck’s theory: if this had happened earlier, one can wonder when some active areas of current mathematics would have been invented.