On the telescope conjecture

Special Algebraic Topology Seminar

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. 


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