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

Convergence of the Motivic Adams Spectral Sequence

“…It is well known ( [13,14]) that in dimension nα, n > 0, π nα (S) is isomorphic to the Witt ring W and that the effect of completion at 2, η in these dimensions is the completion of W at its augmentation ideal. It is easy to show that for general fields, this does not coincide with completion at 2 (one example mentioned in [10] is the field F = Q(x 1 , ..., x m , ...) [i]). Therefore, the second statement of Theorem 1 would be false if we omit the assumption (4).…”

“…Then it is easy to see that there exists an equivalent cell spectrum X with no cells of dimension m + α, m < k. In fact, X can be obtained in ω steps where in i'th step, i ∈ ω, we attach cells in dimension m + i + * α -simply attach to X all cells necessary to cancel homotopy in that dimension, and in the end take homotopy fiber of the canonical map from X -here we are using the fact [13] that (10) π m+ α S = 0 for m < 0.…”

Motivic homotopical Galois extensions

“…This problem was rephrased by Willams [10, p.627] as the question of whether or not KO → KGL hC2 becomes an equivalence after profinite completion. Hu, Kriz and Ormsby in [24,Theorem 20] , they give precise conditions on S under which the map KO → KGL hC2 is an equivalence, even before 2-completion. This result is the key to the proof of the following theorem.…”

“…Our second example is analogous to the classical extension KO → KU . Based on major results of Hu, Kriz and Ormsby in [24] and Berrick, Karoubi, Schlichting and Østvaer [2] on Thomason's homotopy limit problem [39], we state conditions under which the extension from KO, the motivic spectrum representing Hermitian K-theory, to KGL, the motivic spectrum representing algebraic K-theory, is a homotopical C 2 -Galois extension and prove that, in this case, it is faithful on η-complete modules (Theorem 4.3.1).…”

Topological comparison theorems for Bredon motivic cohomology

“…Equivariant motivic homotopy theory was introduced by Voevodsky [Del09] as a tool for understanding symmetric products and motivic Eilenberg-MacLane spaces. Stable equivariant motivic homotopy category was introduced by Hu-Kriz-Ormsby [HKO11] as part of their study of the homotopy limit problem for hermitian Ktheory of fields. In this section we recall definitions and basic results about equivariant motivic homotopy theory.…”