Spaces of embeddings and representations

一个流形N到另一个流形M的嵌入空间具有非常丰富的拓扑，原则上可以用Goodwillie和Weiss的“嵌入演算”来研究。这样的嵌入空间由N的对称群和M的对称群作用，因此它们的上同调给出了这些群的一个有趣的表示来源。设M表示欧几里得空间并由N的对称性作用，通常可以证明这种表示可以分解成相对简单的片断，即所谓的代数表示，但作为N的对称性的表示，这些代数片断可以以有趣的、非代数的方式组合在一起：这使得对N的这种对称群的研究比对它们的相关算术群的研究更丰富。这个项目的目标是发现嵌入空间的上同调是否真的可以以这种方式提供有趣的表示，或者它们是否被约束为实际上是代数的。

The space of embeddings of one manifold N into another manifold M has an extremely rich topology, which can in principle be studied by means of the "embedding calculus" of Goodwillie and Weiss. Such spaces of embeddings are acted upon by the group of symmetries of N and also the group of symmetries of M, and hence their cohomology gives an interesting source of representations of these groups. Letting M denote Euclidean space and acting by symmetries of N, it can often be shown that such representations can be decomposed into relatively simple pieces, so-called algebraic representations, but as representations of the symmetries of N these algebraic pieced can be be combined in interesting, non-algebraic, ways: this makes the study of such groups of symmetries of N richer than the study of their associated arithmetic groups. The goal of this project is to discover whether the cohomology of embedding spaces really can provide interesting representations in this way, or whether they are constrained to actually be algebraic.