A homotopy orbit spectrum for profinite groups

2006/08/10 by Davis, Daniel G., Petrovic, Vojislav
#55P42 #55T25 #Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.math/0608262

Abstract

For a profinite group G, we define an S[[G]]-module to be a certain type of G-spectrum X built from an inverse system \X_i\_i of G-spectra, with each X_i naturally a G/N_i-spectrum, where N_i is an open normal subgroup and G ≅ lim_i G/N_i. We define the homotopy orbit spectrum X_hG and its homotopy orbit spectral sequence. We give results about when its E₂-term satisfies E₂^p,q ≅ lim_i H_p(G/N_i, π_q(X_i)). Our main result is that this occurs if \π_∗(X_i)\_i degreewise consists of compact Hausdorff abelian groups and continuous homomorphisms, with each G/N_i acting continuously on π_q(X_i) for all q. If π_q(X_i) is additionally always profinite, then the E₂-term is the continuous homology of G with coefficients in the graded profinite \widehatℤ[[G]]-module π_∗(X). Other results include theorems about Eilenberg-Mac Lane spectra and about when homotopy orbits preserve weak equivalences.

Related

• Continuous group actions on profinite spaces
• Continuous cohomology and homology of profinite groups
• A lift from group cohomology to spectra for trivial profinite actions
• Profinite equivariant spectra and their tensor-triangular geometry
• Equivariant Eilenberg-Mac Lane spectra in cyclic p-groups