E_k-cells and moduli spaces of manifolds

同调稳定性是研究各种类型的模空间拓扑的一项核心技术。对于流形的模空间，最早是在80年代的S的Heller曲面的情况下建立的；对于高维流形，直到最近几年才取得了进展。在很长一段时间里，有一种或多或少的标准策略来研究同源稳定性，尽管它的细节可以非常具体地用于手头的情况。Galatius，Kupers和Randal-Williams最近的工作利用E_k-代数的概念，给出了一种完全不同的策略来解决这类稳定性问题。它已经应用于许多例子，包括曲面的模空间，但还没有应用到高维流形的模空间。这个项目的目标是在这种情况下发展理论。这个项目自然包括两个部分。在第一部分中，必须证明高维流形的“分裂复形”是高度连通的，这归结为在纯粹的代数环境中展示类似的东西。这里出现的复合体与以前出现过的一些复合体有关，例如在Looijenga和van der Kallen的工作中，但更微妙，需要开发新的技术来处理它们。在第二部分中，我们需要收集关于低次高维流形的模空间的同调群的详细信息，以及它们在同调运算下如何相互作用。这需要一套不同的技术，并将建立在克雷克、克兰尼奇和伯克伦德-哈恩-森格最近的工作基础上。

Homological stability is a central technique in the study of the topology of moduli spaces of all flavours. For moduli spaces of manifolds it was first established in the case of surfaces of Harer in the 80's; for high-dimensional manifolds progress was only made in recent years. For a long time there has been a more-or-less standard strategy for studying homological stability, though its details can be very specific to the case at hand. Recent work of Galatius, Kupers, and Randal-Williams give a completely different strategy to such stability questions, using the notion of E_k-algebras. This has been applied to many examples, including moduli spaces of surfaces, but not yet to moduli spaces of higher-dimensional manifolds. The goal of this project is to develop the theory in this case.This project naturally has two parts. In the first part one must show that the "splitting complex" for high-dimensional manifolds is highly connected, which comes down to showing the analogous thing in a purely algebraic setting. The complexes which arise here are related to some which have appeared before, for example in work of Looijenga and van der Kallen, but are more subtle and new techniques will need to be developed to deal with them. In the second part one needs to amass detailed information about the homology groups of moduli spaces of high-dimensional manifolds in low degree, and how they interact under homology operations. This requires a different set of techniques, and will build on recent work of Kreck, Krannich and Burklund--Hahn--Senger.