Title:
An algebraic model for rational $S^1$-equivariant stable homotopy theory
Author: Brooke Shipley
AMS Classification numbers: 55P62 55P91 55P42 55N91 18E30
Address: Brooke Shipley
1395 Math. Bldg.
Purdue University
West Lafayette, IN 47907 USA
Email addresses: bshipley@math.purdue.edu
Greenlees defined an abelian category $A$ whose derived category is
equivalent to the rational $S^1$-equivariant stable homotopy category
whose objects represent rational $S^1$-equivariant cohomology theories.
We show that in fact the model category of differential graded objects
in $A$ ($dgA$) models the whole rational $S^1$-equivariant stable homotopy
theory. That is, we show that there is a Quillen equivalence between $dgA$
and the model category of rational $S^1$-equivariant spectra, before the
quasi-isomorphisms or stable equivalences have been inverted. This implies
that all of the higher order structures such as mapping spaces, function
spectra and homotopy (co)limits are reflected in the algebraic model.
The new ingredients here are certain Massey product calculations and the
work on rational stable model categories from "Classification of stable
model categories" and "Equivalences of monoidal model categories" with Schwede;
see http://www.math.purdue.edu/~bshipley/