Glen Matthew Wilson: The eta-inverted sphere over the rationals

Abstract: For any field F Morel and and Voevodsky associate to it a stable homotopy category SH(F) that encodes topological information and information about algebraic varieties over F. Just as with the topological stable homotopy category, we would like to calculate the motivic stable homotopy groups of spheres π_{s,w}(1)=SH(F)(Σ^{s,w}1,1). To gain some relevant information about these groups, Guillou and Isaksen—with input from Andrews and Miller—calculated the eta-inverted stable homotopy groups of the 2-complete sphere spectrum over the complex numbers and the real numbers. In this talk we will review the eta-inverted calculations of Guillou and Isaksen and present new results about the structure of the stable homotopy groups of the eta-inverted 2-complete sphere spectrum over fields of cohomological dimension at most 2 (characteristic not 2) and the field of rational numbers.