  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 $K(1)$-local sphere at the prime $2$. As a further
application, we compute and clarify certain power operations in the homotopy of
the $K(1)$-local sphere.

Algebraic Topology;K-Theory and Homology