Morava 𝐾-theories and localisation

The complex oriented cohomology of extended powers

“…Hence by (4.3) n>o it suffices to show that ^(Cp; (Lj^.BP)' 16^) ) = 0. However, by [7] and [8] this will follow by showing that each H 1 (Cp; E(n^(XP)) = 0, which follows by the usual calculations from the hypotheses of the proposition.…”

“…-There are many examples of spaces X satisfying the hypotheses of (4.4). One particularly interesting class of examples is that of spaces X with K{n)°^(X) = 0 for all n since in this case the spectral sequence leading from K(n)*(X) to E(n) (X) collapses and we can appeal to [8]. This class includes the classifying spaces of many finite groups, Eilenberg MacLane spaces, QS 2711 , B0(n), M0(n), BO, ImJ, and so on; see [22] for a detailed discussion of such spaces.…”

I–adic towers in topology

“…by [16,Prop. 7.10], where Γ In (S) is the notation from [13] for the homotopy colimit over a certain type of sequence of finite complexes of type n. The right hand side is isomorphic to…”

“…Only the coaction of the coalgebra F R * (L) is relevant for the E 2 -terms. By comparing explicit descriptions of the Bousfield localization functors L E(n) K(n) and L K(n) from [13] and [16] respectively, we identify the target of (0.6) for T = E(n) and F = K(n) as stated above. We end by considering some examples.…”

Morava Modules and the -Local Picard group

“…The chromatic approach to homotopy theory naturally leads to the study of the K(n)-local stable homotopy category [3,Section 7]. In this thesis we study this category in three different ways.…”