Equivariant algebraic topology and equivariant loop spaces

显然，小的n盘歌剧LD(N)作用于第n个循环空间。这表明n重循环空间(带场系数)的同调是代数算子H_*(LD(N))上的代数。事实上，通常情况下，第n个悬架上的第n个回路具有自由的同调。如果G群在空间上行动，所有这些都要复杂得多。再次清楚的是，LD(V)作用于第V个循环空间。这之后的一切都是复杂的，因为Bredon同调的形式性质，以及与自由循环空间的联系，并且因为它可能是最好的考虑RO(G)-分级。May，Hill，Wilson等人在小循环2-群上的工作表明，它存在着许多有趣的结构，但仍有许多问题没有得到解答。这个项目的目的是了解LD(V)的同调的代数运算，进而了解当群是小的循环2-群时，小V的Vth圈空间的同调。

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.