Coloured topological operads and moduli spaces of surfaces with multiple boundary curves

Abstract

While it is a classical result that the collection of moduli spaces of surfaces with a single boundary curve is an <nobr>E₂-algebra</nobr> (more precisely: it admits an action of the little 2-cubes operad C₂), we need a coloured version of C₂ which understands a cluster of squares as a single input with a certain multiplicity, if we want to establish an action on the collection of moduli spaces of surfaces with multiple boundary curves in a similar way. <br /> Moreover, Bödigheimer introduced a finite multisimplicial model for moduli spaces, which is useful for explicit homological calculations. In order to construct an operadic action on this specific model, we have to additionally require a certain coupling behaviour among squares belonging to the same input. This gives rise to a family of suboperads, called <em>vertical operads</em>. <br /> We analyse these operads from several perspectives: on the one hand, their operation spaces and free algebras are modelled by clustered and vertical configuration spaces, whose homology, homological stability, and iterated bar constructions we investigate in the first chapters. On the other hand, we study the homotopy theory and the homology of their algebras and use the arising operations to describe the unstable homology of moduli spaces. <br /> Finally, it turns out that the developed methods are also useful to solve a problem of a seemingly different flavour: for a fixed space A, the collection of moduli spaces of surfaces <em>parametrised</em> over A is itself an <nobr>E₂-algebra,</nobr> and its group completion is an infinite loop space. We identify the underlying spectrum in the spirit of Madsen and Weiss.