---
tag: 68650624FF
kind: counterexample
name: Telescope conjecture
msc: [55P42, 55Q51, 55P60]
uses: [B8XP2K7RS1, 300V40FC1B, 1GHF4EKBXS]
sources:
  - url: https://en.wikipedia.org/wiki/Ravenel's_conjectures
---

## Counterexample

Fix a prime $p$ and $n \ge 1$.
Let $X$ be a finite $p$-local [[spectrum|1GHF4EKBXS]] of type $n$.
Let $v \colon \Sigma^d X \to X$ be a $v_n$ self-map, as given by the periodicity theorem among the [[Ravenel conjectures|B8XP2K7RS1]].
The telescope of $v$ is the homotopy colimit
$$v^{-1} X = \operatorname{hocolim}\left( X \xrightarrow{v} \Sigma^{-d} X \xrightarrow{v} \Sigma^{-2d} X \to \cdots \right).$$
The telescope conjecture asserted that the natural map $v^{-1} X \to L_{K(n)} X$ is an equivalence, where $L_{K(n)}$ is Bousfield localization at the $n$-th Morava K-theory.
The conjecture is false for every prime $p$ and every height $n \ge 2$.
In 2023 Burklund, Hahn, Levy and Schlank showed that for such $p$ and $n$ the map $v^{-1} X \to L_{K(n)} X$ is not an equivalence.

## Remarks

The telescope $v^{-1} X$ does not depend on the choice of $v$, and its Bousfield class depends only on $n$; the localization at it is written $L_{T(n)}$.
So the conjecture says $L_{T(n)} = L_{K(n)}$.
At height $1$ the conjecture is true: Mahowald proved it for $p = 2$ and Miller for $p$ odd.
The counterexample of Burklund, Hahn, Levy and Schlank uses algebraic K-theory to build $T(n)$-local spectra that are not $K(n)$-local.
It shows that the homotopy of telescopes is genuinely larger than what $K(n)$-local computations see.

## Explanations

### L1 human

A finite spectrum of height $n$ has a periodic self-map, and inverting it gives a "telescope".
Ravenel conjectured that the telescope is the same as the localization at Morava K-theory, which is much easier to compute.
This is true at heights $0$ and $1$.
At every height from $2$ on it is false, so the telescope holds information that Morava K-theory does not see.