Topological moduli spaces, homological stability, and operads

This thesis consists of two relatively independent parts. In the first part, we study operads with homological stability, which are topological operads that satisfy a homological stability condition. Despite the fact that these operads are in general very far from being E∞-operads, their algebras give rise to infinite loop spaces upon group completion. We show that under a mild condition on an operad with homological stability, one can naturally associate topologically enriched symmetric strict monoidal categories to its algebras such that the group completion of the algebra is weakly equivalent to the loop space on the classifying space of the associated category, which is evidently an infinite loop space. We also show that this new infinite loop space structure coming from the associated category is equivalent to the previously known infinite loop space structure on the group completion of the algebra. The second part is concerned with quotients of mapping class groups Γg,1 of oriented surfaces with one boundary component by the subgroups ϕg,1(k) in their Johnson filtration. We show that the stable classifying spaces Z × B (Γ∞/ϕ∞(k))+ after Wuillen's plus-construction are infinite loop spaces, fitting into a tower of infinite loop space maps that interpolates between the infinite loop spaces Z × BΓ+∞ and Z × B (Γ∞/ϕ∞(1))+ ≃ Z × BSp(Z)+ . We also show that for each level k in the Johnson filtration, the homology of these quotients with suitable systems of twisted coeficients stabilises as the genus g of the surface tends to infinity.