Title: The Spanier-Whitehead duals of some higher real K-theory spectra
Abstract: Higher real K-theories are central objects in chromatic homotopy theory. Their homotopy groups approximate the K(n)-local part of the stable homotopy groups of spheres, moreover the theory of finite resolutions suggests that these spectra and their Spanier-Whitehead duals form building blocks for the K(n)-local sphere. Work of Beaudry-Goerss-Hopkins-Stojanoska shows that taking the dual of one of these theories amounts to a twist by a certain representation sphere. Interestingly, in every case known to the speaker a higher real K-theory ends up being self-dual up to some suspension.
In this talk we will begin with an introduction to the ideas outlined above then show how to identify the representation sphere twist in some cases, working always at the prime 2. Then we make use of recent advances in equivariant computational techniques stemming from Hill-Hopkins-Ravenel's work on the Kervaire invariant problem to identify an integer suspension shift at some low heights.
