Equivariant algebraic topology and equivariant loop spaces

It is clear that the little n-discs operad LD(n) acts on an nth loop space. This shows that the homology of an n-fold loop space (with field coefficients) is an algebra over the algebraic operad H_*(LD(n)). Indeed, it is often true that the nth loops on an nth suspension has homology which is free. All of this is much more complicated if a group G acts on the spaces. It is once again clear that LD(V) acts on a Vth loop space. Everything after this is complicated, because of the formal properties of Bredon homology, and the connection with free loop spaces, and because it may be best to take into account the RO(G)-grading. There is work of May, Hill, Wilson and others on small cyclic 2-groups, which shows that there is a great deal of interesting structure, but which leaves many questions unanswered. The project aims to understand the algebraic operad of the homology of LD(V), and then homology of Vth loop spaces for small V when the group is a small cyclic 2-group.