Ravenel conjectures
Encyclopedia
The Ravenel conjectures are a set of mathematical conjectures in the field of stable homotopy theory
posed by Douglas Ravenel
at the end of a paper published in 1984. It was earlier circulated in preprint. The problems involved have largely been resolved, with all but the "telescope conjecture" being proved in later papers by others. The telescope conjecture is now generally believed not to be true, though there are some conflicting claims concerning it in the published literature, and is taken to be an open problem. Ravenel's conjectures exerted influence on the field through the founding of the approach of chromatic homotopy theory.
The first of the seven conjectures, then the nilpotence conjecture, is now the nilpotence theorem
. The telescope conjecture, which was #4 on the original list, remains of substantial interest because of its connection with the convergence of an Adams–Novikov spectral sequence. While opinion is against the truth of the original statement, investigations of associated phenomena (for a triangulated category
in general) have become a research area in its own right.
Stable homotopy theory
In mathematics, stable homotopy theory is that part of homotopy theory concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor...
posed by Douglas Ravenel
Douglas Ravenel
Douglas Conner Ravenel is an American mathematician known for work in algebraic topology.-Life:He received his Ph.D. from Brandeis University in 1972 under the direction of Edgar H. Brown, Jr. with a thesis on exotic characteristic classes of spherical fibrations. From 1971 to 1973 he was...
at the end of a paper published in 1984. It was earlier circulated in preprint. The problems involved have largely been resolved, with all but the "telescope conjecture" being proved in later papers by others. The telescope conjecture is now generally believed not to be true, though there are some conflicting claims concerning it in the published literature, and is taken to be an open problem. Ravenel's conjectures exerted influence on the field through the founding of the approach of chromatic homotopy theory.
The first of the seven conjectures, then the nilpotence conjecture, is now the nilpotence theorem
Nilpotence theorem
In algebraic topology, the nilpotence theorem gives a condition for an element of the coefficient ring of a ring spectrum to be nilpotent, in terms of complex cobordism. It was conjectured by and proved by .-Nishida's theorem:...
. The telescope conjecture, which was #4 on the original list, remains of substantial interest because of its connection with the convergence of an Adams–Novikov spectral sequence. While opinion is against the truth of the original statement, investigations of associated phenomena (for a triangulated category
Triangulated category
A triangulated category is a mathematical category satisfying some axioms that are based on the properties of the homotopy category of spectra, and the derived category of an abelian category. A t-category is a triangulated category with a t-structure.- History :The notion of a derived category...
in general) have become a research area in its own right.