On the Brown Comenetz dual of the K(2)-local sphere at the prime 2

Hans-Werner Henn, Institut de Recherche Mathématique Avancée, Université de Strasbourg

Zoom link: https://princeton.zoom.us/j/96282936122

Passcode: 998749

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 I_n 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 I_n 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 I_n 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 then 3 is sufficiently large. The case p=3 had been settled in joint work with Paul Goerss several years ago. 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 2^9. We use chromatic splitting in order to get information on the error term.