Title: Leibniz Formulas for Cyclic Homotopy Fixed Point Spectra
Authors: Robert R. Bruner and John Rognes
MSC-class:  19D55, 55P43, 55P91, 55S12, 55T05.
ArXiv ID: math.AT/0406081
Addresses:

Robert R. Bruner 
   Department of Mathematics
   Wayne State University
   Detroit, Michigan 48067
   USA
rrb@math.wayne.edu

John Rognes
   Department of Mathematics
   University of Oslo
   Box 1053, Blindern
   NO-0316 Oslo
   Norway
rognes@math.uio.no

Abstract:
We analyze the homotopy fixed point spectrum of a circle-equivariant
commutative S-algebra R in homological terms.  There is a homological
homotopy fixed point spectral sequence that converges conditionally to
the continuous homology of the homotopy fixed point spectrum.  We show
that there are Dyer-Lashof operations Q^i acting on this algebra spectral
sequence, and that its differentials are completely determined by those
originating on the vertical axis. More surprisingly, we show that for each
class x in the E^{2r}-term of the spectral sequence there are 2r other
classes in the E^{2r}-term (obtained mostly by Dyer-Lashof operations
on x) that are infinite cycles, i.e., survive to the E^infty-term. We
apply this to completely determine the differentials in the homological
homotopy fixed point spectral sequences for the topological Hochschild
homology spectra R = THH(B) of many S-algebras, including B = MU, BP,
ku, ko and tmf. Similar results apply for all finite subgroups of the
circle, and for the Tate- and homotopy orbit spectra. This work is part
of a homological approach to calculating topological cyclic homology
and algebraic K-theory of commutative S-algebras.