The two definitions of the index difference

Given two metrics of positive scalar curvature metrics on a closed spin manifold, there is a secondary index invariant in real $K$-theory. There exist two definitions of this invariant, one of homotopical flavour, the other one defined by a index problem of Atiyah-Patodi-Singer type. We give a complete and detailed proof of the folklore result that both constructions yield the same answer. Moreover, we generalize this to the case of two families of positive scalar curvature metrics, parametrized by a compact space. In essence, we prove a generalization of the classical "spectral-flow-index theorem" to the case of families of real operators.Comment: Revised versio

On the homotopy type of the Deligne-Mumford compactification

An old theorem of Charney and Lee says that the classifying space of the category of stable nodal topological surfaces and isotopy classes of degenerations has the same rational homology as the Deligne-Mumford compactification. We give an integral refinement: the classifying space of the Charney-Lee category actually has the same homotopy type as the moduli stack of stable curves, and the etale homotopy type of the moduli stack is equivalent to the profinite completion of the classifying space of the Charney-Lee category.Comment: 14 pages, published versio

Generalised Miller-Morita-Mumford classes for block bundles and topological bundles

The most basic characteristic classes of smooth fibre bundles are the generalised Miller-Morita-Mumford classes, obtained by fibre integrating characteristic classes of the vertical tangent bundle. In this note we show that they may be defined for more general families of manifolds than smooth fibre bundles: smooth block bundles and topological fibre bundles.Comment: 18 page