The chromatic splitting conjecture (CSC) is an important open problem in stable homotopy theory. Although Beaudry has shown that the strongest version fails when n=p=2, one can still hope that the conjecture is valid at large primes, or up to a filtration that is nearly split in some appropriate sense.
The CSC gives a description of Ln-1LK(n)(S), but from that one can deduce similar descriptions of many other spectra, including those of the form LK(n1) LK(n2) … LK(nr)(S) with n1 < … < nr. The spectrum Ln(S) can be expressed as the homotopy inverse limit of a diagram of spectra of that form, and one can ask whether that, and various similar phenomena, are consistent with the CSC. We will explain a conjecture about how all this fits together, which involves some interesting algebra and combinatorics. We will also explain how Morava K-theory Euler characteristics can be used to do some basic consistency checks.
This video is part of the New Directions in Group Theory and Triangulated Categories seminar series.
