Ravenel's conjectures explained

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.^[1] It was earlier circulated in preprint.^[2] The problems involved have largely been resolved, with all but the "telescope conjecture" being proved in later papers by others.^{[3] [4] [2]} 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, was proved in 1988 and is now known as 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 has been generally 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.^{[5] [6]}

On June 6, 2023, Robert Burklund, Jeremy Hahn, Ishan Levy, and Tomer Schlank announced a disproof of the telescope conjecture.^[7] Their preprint is submitted to the arXiv on October 26, 2023.^[8]

See also

• Homotopy groups of spheres

Notes and References

1. Localization with Respect to Certain Periodic Homology Theories . Douglas C. . Ravenel . American Journal of Mathematics . 106 . 2 . 1984 . 351–414 . 2374308 . 0737778 . 10.2307/2374308.
2. The mathematical work of Douglas C. Ravenel . Homology, Homotopy and Applications . 10 . 2008 . 3, Proceedings of a Conference in Honor of Douglas C. Ravenel and W. Stephen Wilson . 1–13 . Hopkins . Michael J.. 10.4310/HHA.2008.v10.n3.a1 . free .
3. Devinatz . Ethan S. . Hopkins . Michael J. . Michael J. Hopkins. Smith . Jeffrey H. . Nilpotence and stable homotopy theory. I . 10.2307/1971440 . 960945 . 1988 . . Second Series . 0003-486X . 128 . 2 . 207–241. 1971440 .
4. 120991. Nilpotence and Stable Homotopy Theory II . Hopkins . Michael J. . Michael J. Hopkins. Smith . Jeffrey H. . 1998 . . Second Series . 148 . 1 . 1–49. 10.2307/120991 . 10.1.1.568.9148 .
5. Brüning . Kristian . Subcategories of Triangulated Categories and the Smashing Conjecture . 25. Dissertation zur Erlangung des akademischen Grades. 2007.
6. Jack. Hall. David. Rydh. 2016-06-27. The telescope conjecture for algebraic stacks. Journal of Topology. 10. 3. 776–794. 1606.08413. en. 10.1112/topo.12021. 119336098 .
7. Web site: Hartnett . Kevin . 2023-08-22 . An Old Conjecture Falls, Making Spheres a Lot More Complicated . 2023-08-22 . Quanta Magazine . en.
8. Web site: Burklund . Robert . Hahn . Jeremy . Levy . Ishan . Schlank . Tomer . K-theoretic counterexamples to Ravenel's telescope conjecture . 27 October 2023.