University of David
Another achievement of this golden period concerned the complex cobordismring and its relation to the theory of formal groups. This idea is the basis of most recent work in stable homotopy theory, beginning with the determination by Hopkins of the primes of the stable homotopy category and the “chromatic” picture of the homotopy groups of spheres. Milnor’s calculation of the complex cobordism ring in 1960 by means of the Adams spectral sequence had been one of the triumphs of algebraic topology. Quillen had been thinking about Grothendieck’s theory of “motives” as a universal cohomology theory in algebraic geometry and also about the use Grothendieck had made of bundles of projective spaces in his earlier work on Chern classes and the Riemann-Roch theorem. He saw that complex cobordism had a similar universal role among those cohomology theories for smooth manifolds in which vector bundles have Chern classes, and that the fundamental invariant of such a theory is the formal group law which describes how the first Chern class of a line bundle behaves under the tensor product. He made the brilliant observation that the complex cobordism ring is the base of the universal formal group, and he succeeded in devising a completely new calculation of it, not using the Adams spectral sequence, but appealing instead to the fundamental properties of the geometric power operations on manifolds. This work is yet another mélange of Grothendieck-style ideas with more concrete and traditional algebraic topology. After his one amazing paper on this subject he seems never to have returned to the area.