Equivariant algebraic topology and equivariant loop spaces

等变代数拓扑和等变环空间

• 批准号: 2105861
• 金额: --
• 依托单位: University of Warwick
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2018
• 资助国家: 英国
• 起止时间: 2018 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2105861
• 关键词: Equivariant; algebraic; topology; equivariant; loop

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.

很明显，小的n圆盘操作数LD（n）作用于第n个循环空间。这表明n重圈空间（具有场系数）的同调是代数算子H *（LD（n））上的代数。实际上，在第n个悬挂上的第n个环具有自由的同调，这常常是真的。如果群G作用在空间上，所有这些都要复杂得多。再次清楚的是，LD（V）作用于Vth循环空间。由于Bredon同调的形式性质和与自由循环空间的联系，以及考虑RO（G）-分级可能是最好的，因此在此之后的一切都是复杂的。有工作的五月，希尔，威尔逊和其他人的小循环2群，这表明，有大量的有趣的结构，但留下了许多问题没有答案。该项目旨在了解LD（V）的同调的代数运算，然后当群是小循环2-群时，小V的Vth循环空间的同调。