Stable moduli spaces of manifolds - Randal-Williams, O (2009)
In this thesis we make several contributions to the theory of moduli spaces of smooth manifolds, especially in dimension two. In Chapter 2 (joint with Soren Galatius) we give a new geometric proof of a generalisation of the Madsen-Weiss theorem, which does not rely on the tangential structure under investigation having homological stability. This allows us to compute the stable homology of moduli spaces of surfaces equipped with many different tangential structures. In Chapter 3 we give a general approach to homological stability problems, especially focused on stability for moduli spaces of surfaces with tangential structure. We give a sufficient condition for a structure to exhibit homological stability, and thus obtain stability ranges for many tangential structures of current interest (orientations, maps to a simply-connected background space, etc.), which match or improve the previously known ranges in all cases. In Chapter 4 we define and study the cobordism category of submanifolds of a fixed background manifold, and extend the work of Galatius-Madsen-Tillmann-Weiss to identify the homotopy type of these categories. We describe several applications of this theory. In Chapter 5 we compute the stable (co)homology of the non-orientable mapping class group, and find a family of geometrically-defined torsion cohomology classes. This is in contrast to the oriented mapping class group, where few are known. In Chapter 6 (joint with Johannes Ebert) we study the divisibility of certain characteristic classes of bundles of unoriented surfaces introduced by Wahl, analogues of the Miller-Morita-Mumford classes for unoriented surfaces. We show them to be indivisible in the free quotient of cohomology.