In
mathematics, 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 In mathematics, chromatic homotopy theory is a subfield of stable homotopy theory that studies complex-oriented cohomology theories from the "chromatic" point of view, which is based on Quillen's work relating cohomology theories to formal groups. ...
.
The first of the seven conjectures, then the ''nilpotence conjecture'', was proved in 1988 and is now known as the nilpotence theorem
In algebraic topology, the nilpotence theorem gives a condition for an element in the homotopy groups of a ring spectrum to be nilpotent, in terms of the complex cobordism spectrum \mathrm. More precisely, it states that for any ring spectrum R, t ...
.
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 In mathematics, the Adams spectral sequence is a spectral sequence introduced by which computes the stable homotopy groups of topological spaces. Like all spectral sequences, it is a computational tool; it relates homology theory to what is now c ...
. While opinion has been generally against the truth of the original statement, investigations of associated phenomena (for a triangulated category In mathematics, a triangulated category is a category with the additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the derived category of an abelian category, as well as the stable homotopy cat ...
in general) have become a research area in its own right.
On June 6, 2023, Robert Burklund, Jeremy Hahn, Ishan Levy, and Tomer Schlank announced a disproof of the telescope conjecture. Their preprint is submitted on October 26, 2023.
See also
* Homotopy groups of spheres
In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure o ...
References
{{Reflist
Homotopy theory
Conjectures