Michael J. Hopkins - Work - The Ravenel Conjectures

The Ravenel Conjectures

For more details on this topic, see Ravenel conjectures.

The Ravenel conjectures very roughly say: complex cobordism (and its variants) see more in the stable homotopy category than you might think. For example, the nilpotence conjecture states that some suspension of some iteration of a map between finite CW-complexes is null-homotopic iff it is zero in complex cobordism. This was proven by Devinatz, Hopkins and Jeff Smith (published in 1988). The rest of the Ravenel conjectures (except for the telescope conjecture) were proven by Hopkins and Smith soon after (published in 1998). Another result in this spirit proven by Hopkins and Ravenel is the chromatic convergence theorem, which states that one can recover a finite CW-complex from its localizations with respect to wedges of Morava K-theories.