 

Daniel G. Davis
Homotopy fixed points for profinite groups emulate homotopy fixed points for discrete groups view print


Published: 
November 24, 2013 
Keywords: 
Homotopy fixed point spectrum, discrete Gspectrum 
Subject: 
55P42, 55P91 


Abstract
If K is a discrete group and Z is a Kspectrum, then
the homotopy fixed point spectrum
Z^{hK} is Map_{∗}(EK_{+}, Z)^{K}, the
fixed points of a familiar expression. Similarly, if G is a
profinite group and X is a discrete Gspectrum,
then X^{hG} is often given by
(H_{G,X})^{G}, where
H_{G,X} is a
certain explicit construction given by a
homotopy limit in the category of discrete Gspectra.
Thus, in each of two common equivariant settings,
the homotopy fixed point
spectrum is equal to the fixed points of an explicit
object in the ambient equivariant category.
We enrich this pattern by proving in a precise sense
that the discrete Gspectrum
H_{G,X} is
just "a profinite version" of Map_{∗}(EK_{+}, Z): at each
stage of its construction, H_{G,X}
replicates in the setting of discrete Gspectra
the corresponding stage in the formation of Map_{∗}(EK_{+}, Z)
(up to a certain natural identification).


Author information
Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA 70504, U.S.A.
dgdavis@louisiana.edu

