nLab telescope conjecture

Idea

In the stable homotopy category of $p$-local spectra for some prime number $p$, there are two collections of similarly behaved spectra: the Morava K-theories and spectra which are the telescopes of finite spectra under their $v_n$-maps. The former are usually denoted $K(n)$ and the latter are often denoted $T(n)$. The telescope conjecture is that localization at $K(n)$ and localization at $T(n)$ are (Bousfield) equivalent.

The conjecture is known to be true for $n=1$. A disproof of this conjecture for $n\geq 2$ using algebraic K-theory has been released by Burklund, Hahn, Levy & Schlank 2023.

References

Proof at height 1:

• Mark Mahowald, bo-Resolutions (1981)

• Haynes Miller, On relations between Adams spectral sequences, with an application to the stable homotopy of a Moore space (1981)

Disproof at height $\geq 2$:

• Robert Burklund, Jeremy Hahn, Ishan Levy, Tomer Schlank, $K$-theoretic counterexamples to Ravenel’s telescope conjecture, (arXiv:2310.17459)