The Brown Comenetz dual I of the sphere represents the functor which on a spectrum X is given by the Pontryagin dual of the 0-th homotopy group of X. For a prime p and a chromatic level n there is a K(n)-local version In of I. For a type n-complex X, this is given by the Pontryagin dual of the 0-th homotopy group of the K(n)-localization of X. By work of Hopkins and Gross, the homotopy type of the spectra In for a prime p is determined by its Morava module if p is sufficiently large. For small primes, the result of Hopkins and Gross determines In modulo an “error term”. For n=1 every odd prime is sufficiently large and the case of the prime 2 has been understood for almost 30 years. For n>2 very little is known if the prime is small. For n=2 every prime bigger than 3 is sufficiently large. The case p=3 has been settled in joint work with Paul Goerss. This talk is a report on work in progress with Paul Goerss on the case p=2. The “error term” is given by an element in the exotic Picard group which in this case is an explicitly known abelian group of order 29. We use chromatic splitting in order to get information on the error term.
This video is part of the New Directions in Group Theory and Triangulated Categories seminar series.
