Author: J.P.C.Greenlees

Abstract: We make a systematic study of rational $S^1$-equivariant 
cohomology theories, or rather of their representing objects, 
rational $S^1$-spectra.

In   Part I we construct a complete  algebraic model for the homotopy 
category of $S^1$-spectra, reminiscent of the localization theorem. 
The model is of homological dimension one, and 
simple enough to allow practical calculations; in particular we obtain 
a classification of rational $S^1$-equivariant cohomology theories.
The model for semifree spectra is the derived category of 
the abelian category whose objects are $\Q [c]$-modules $N$
with a graded vector space $V$ giving an isomorphism 
$N[c^{-1}] = \Q [c , c^{-1}] \tensor V$; the model for arbitrary 
spectra is an appropriate generalization.  
 
In  Part II we identify the algebraic counterparts of all the usual $S^1$-spectra and  constructions on $S^1$-spectra.  This enables us 
in Part III to give a rational analysis of a number of interesting 
phenomena, such as the Atiyah-Hirzebruch spectral sequence, the Segal 
conjecture, $K$-theory and topological cyclic cohomology. 

Status: Preprint