Monoidal algebraic models for rational S1 equivariant ring spectra
Cohomology theories are of great importance for studying topological spaces. They take as input topological spaces and as output give a collection of abelian groups. They satisfy a useful collection of axioms which (in theory) make these groups computable.
If X is a space with an action of a compact Lie group G and E is a cohomology theory, then E*(X) also has a G-action. But this doesn't usually tell us a great deal about the action, for example if G is the circle group S1, then the action on cohomology is always trivial. So there is a need for cohomology theories that use the G-action in a more fundamental way.
A purely algebraic category that describes rational S1-equivariant cohomology theories (or more accurately rational s1 spectra) was constructed by Greenlees. In this talk I will introduce this category and explain how it can be used to model rational S1-equivariant cohomology theories with a multiplication (also called a cup product).