Nilpotence and Periodicity in Stable Homotopy Theory. (AM-128)

Bousfield localization functors and Hopkins’ chromatic splitting conjecture

“…Recall that a finite spectrum X has type n if K(i) * (X) = 0 for i < n but K(n) * (X) = 0. Every finite spectrum is of some finite type, and the periodicity theorem of J. Smith, written up in [Rav92], says that there is a spectrum of type n for all n. Let C n denote the class of all finite spectra of type at least n. Then [HS] any nonempty collection of finite spectra that is closed under cofibrations and retracts is some C n .…”

“…First note that because L f n and L n are both smashing (see [Rav92] for L n ), so is A n . That is, A n X = A n S 0 ∧ X for all X.…”

“…Hopkins and Ravenel later extended this to the chromatic convergence theorem [Rav92]. If we denote, as usual, the localization with respect to the first n + 1 Morava K-theories K(0) ∨ · · · ∨ K(n) by L n , the chromatic convergence theorem says that for finite X, the tower .…”

Transfer maps for fibrations

“…If R = S 0 and F is stably finite, then F is R-small and χ R (F ) ∈ π 0 R = Z is the ordinary Euler characteristic χ(F ) (see [3, 2.1]). If R is the Eilenberg-MacLane spectrum H Z/p or H Q, a Morava K-theory spectrum K(n) [12], or mod p complex K-theory then F is R-small as long as π * (R ∧ F + ) is finitely generated as a module over π * R. This follows from the Kunneth theorem for these theories [12, p. 175]; in fact, in all of these cases there are evident isomorphisms…”

Completions in Algebra and Topology

“…We complete the general theory with an easy, but tantalizing, result that will specialize to give part of the proof of the Chromatic Convergence Theorem of HopkinsRavenel [23]. It well illustrates how the algebraic information in Section 2 can have non-obvious topological implications.…”