A Lyndon-Hochschild-Serre spectral sequence for certain homotopy fixed point spectra

Abstract

Abstract. Let H and K be closed subgroups of the extended Morava stabilizer group Gn and suppose that H is normal in K. We construct a strongly convergent spectral sequence H *

Homotopy groups of homotopy fixed point spectra associated to En

“…(As usual, DX denotes the Spanier-Whitehead dual of X .) There is, however, no integer α for which (3)(4)(5)(6)(7)(8)(9)(10) is satisfied for all X . This contrasts with the situation when n = 1: here we have I 1 Σ 2 L 1 (S 0 p ) (if p > 2), where S 0 p denotes S 0 completed at p, and thus F(X, I 1 ) Σ 2 L 1 DX whenever X is a rationally acyclic finite spectrum.…”

“…Suppose there existed an integer c with (3)(4)(5)(6)(7)(8)(9)(10)(11) F(M(p, v k 1 ), I 2 ) Σ c L 2 DM(p, v k 1 ) for a cofinal set of k, where M(p, v k 1 ) denotes a finite spectrum with BP * M(p, v k 1 ) = BP * /(p, v k 1 ). In addition, we may assume that DM(p, v k 1 ) Σ −2k(p−1)−2 M(p, v k 1 ).…”

“…(This follows from the fact that any p-adic analytic profinite group is of type p − FP ∞ in the language of Symonds and Weigel [20].) But, in an earlier paper [4], we constructed a strongly convergent spectral sequence…”

“…(See Strickland [19,Proposition 17] or Devinatz [2] for p ≥ 5; note, however, that we are using Strickland's equivalent definition of E 2 * M(p, v k 1 ) ∼ .) Then (3)(4)(5)(6)(7)(8)(9)(10)(11) implies that…”

An Orientation Map for Height p-1 Real E Theory

“…This means that any sphere bundle that comes from a complex vector bundle with vanishing first Chern class is -orientable. Similarly, the map [6] → GL 1 ( ) is -orientable so any sphere bundle that comes from a complex vector bundle with vanishing first two Chern classes is -orientable. The localizations (1) and (2) are the = 2 and = 3 cases of a family of cohomology theories called higher real -theories −1 .…”