E_k-cells and moduli spaces of manifolds

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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.