Title: On the Brown Comenetz dual of the K(2)-local sphere at the prime 2 - a sequel to the conference talk in March 2023 (joint work in progress with Paul Goerss)
Speaker: Hans-Werner Henn
Speaker Info: University of Strasbourg
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 the homotopy groups of the function spectrum F(X,I_n) are given by the Pontryagin-dual of the homotopy groups of the K(n)-localization of X.Date: Monday, May 08, 2023
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 with respect to n and this Morava module has been determined. For small primes the result of Hopkins and Gross determines I_n modulo an “error term”.
This talk is a report on work in progress with Paul Goerss on the case n=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 first use chromatic splitting in order to narrow down the choices for the error term and then use specific information about the homotopy groups of a particular finite type 2-complex to nail down the error term.