# On the telescope conjecture

# On the telescope conjecture

The organizers for this **online seminar** are Tony Bahri, Rider University, Martin Bendersky, CUNY and Princeton University, Bill Browder, Princeton University, and Doug Ravenel, University of Rochester.

*SCHEDULE*

8:20 - 8:30 **Welcome**, William Browder

8:30 - 9:00 **Introduction**, Douglas Ravenel

9:15 - 10:15 **An overview of the proof. **Ishan Levy, Massachusetts Institute of Technology

Ravenel’s telescope conjecture asserts that the T(n)-local categories, which detect the v_n-periodic part of the stable homotopy groups of spheres, agree with the K(n)-local categories, which are computationally accessible via the moduli of height n formal groups. With Burklund, Hahn, and Schlank, we prove that this conjecture is false for n>=2, by constructing counterexamples using algebraic K-theory. I will give an overview of the proof for n=2, where we show the T(2)-local algebraic K-theory of the K(1)-local sphere is not K(2)-local.

10:30 - 11:30** Boundedness of Cyclotomic Spectra. **Tomer Schlank,The Hebrew University of Jerusalem

A key step in the disproof of the telescope conjecture is proving that certain cyclotomic spectra are bounded in the Antieau-Nikolaus $t$-structure. In this talk we shall review the category of cyclotomic spectra and certain boundedness and compactness properties a cyclotomic spectrum can possess.

11:30 - 1:00 BREAK

1:00 - 1:30 **Discussion.** Douglas Ravenel, Moderator

1:45 - 2:45 **Hochschild homology of fixed points by unipotent Z-actions.**** **Jeremy Hahn, Massachusetts Institute of Technology

If R is a ring with a locally unipotent Z-action, then I will discuss how to understand THH(R^{hZ}) in terms of THH(R)^{hZ}. Special attention will be given to the sphere spectrum with trivial Z-action, and to topological complex K-theory with Z-action generated by an Adams operation. We do this with an eye toward checking whether Lichtenbaum—Quillen properties are preserved upon taking fixed points by locally unipotent Z-actions.

3:00 - 4:00 **Asymptotic constancy for THH. **Robert Burklund, University of Copenhagen

Given a connective, height n ring spectrum R with a locally unipotent Z-action, it turns out that under strong finiteness assumptions there are isomorphisms of cyclotomic spectra V \otimes THH(R^{hp^kZ}) \cong V \otimes THH(R^{BZ})for V finite of type n+2 and k>>0. As a consequence, in many cases of interest we may replace a non-trivial Z action by a trivial one when computing TC. Using these ideas I will then present the final details in our disproof of the telescope conjecture.

4:00 - 4:10 **Closing comments.** Martin Bendersky