Daniel G. Davis
Homotopy fixed points for profinite groups emulate homotopy fixed points for discrete groups
||November 24, 2013
||Homotopy fixed point spectrum, discrete G-spectrum
If K is a discrete group and Z is a K-spectrum, then
the homotopy fixed point spectrum
ZhK is Map∗(EK+, Z)K, the
fixed points of a familiar expression. Similarly, if G is a
profinite group and X is a discrete G-spectrum,
then XhG is often given by
HG,X is a
certain explicit construction given by a
homotopy limit in the category of discrete G-spectra.
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 G-spectrum
just "a profinite version" of Map∗(EK+, Z): at each
stage of its construction, HG,X
replicates in the setting of discrete G-spectra
the corresponding stage in the formation of Map∗(EK+, Z)
(up to a certain natural identification).
Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA 70504, U.S.A.