The homotopy limit problem for Hermitian K-theory, equivariant motivic homotopy theory and motivic Real cobordism

The homotopy limit problem for Karoubiʼs Hermitian K-theory (Karoubi, 1980) [26], was posed by Thomason (1983) [44]. There is a canonical map from algebraic Hermitian K-theory to the Z/2-homotopy fixed points of algebraic K-theory. The problem asks, roughly, how close this map is to being an isomorphism, specifically after completion at 2. In this paper, we solve this problem completely for fields of characteristic 0 (Theorems 16, 20). We show that the 2-completed map is an isomorphism for fields F of characteristic 0 which satisfy cd2(F[i])<∞, but not in general.