Higher semiadditive Grothendieck-Witt theory and the 𝐾(1)-local sphere

Abstract

We develop a higher semiadditive version of Grothendieck-Witt theory. We then apply the theory in the case of a finite field to study the higher semiadditive structure of the K ( 1 ) K(1) -local sphere S K ( 1 ) \mathbb {S}_{K(1)} at the prime 2 2 , in particular realizing the non- 2 2 -adic rational element 1 + ε ∈ π 0 S K ( 1 ) 1+\varepsilon \in \pi _0\mathbb {S}_{K(1)} as a “semiadditive cardinality.” As a further application, we compute and clarify certain power operations in π 0 S K ( 1 ) \pi _0\mathbb {S}_{K(1)} .