Johannes Ebert

Infinite loop spaces and positive scalar curvature

We study the homotopy type of the space of metrics of positive scalar curvature on high-dimensional compact spin manifolds. Hitchin used the fact that there are no harmonic spinors on a manifold with positive scalar curvature to construct a secondary index map from the space of positive scalar metrics to a suitable space from the real K-theory spectrum. Our main results concern the nontriviality of this map. We prove that for

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

2n \ge 6

On the homotopy type of the Deligne–Mumford compactification

2n≥6, the natural KO-orientation from the infinite loop space of the Madsen–Tillmann–Weiss spectrum factors (up to homotopy) through the space of metrics of positive scalar curvature on any 2n-dimensional spin manifold. For manifolds of odd dimension

Pontrjagin–Thom maps and the homology of the moduli stack of stable curves

2n+1 \ge 7

Characteristic Classes of Spin Surface Bundles: Applications of the Madsen-Weiss Theory

2n+1≥7, we prove the existence of a similar factorisation. When combined with computational methods from homotopy theory, these results have strong implications. For example, the secondary index map is surjective on all rational homotopy groups. We also present more refined calculations concerning integral homotopy groups. To prove our results we use three major sets of technical tools and results. The first set of tools comes from Riemannian geometry: we use a parameterised version of the Gromov–Lawson surgery technique which allows us to apply homotopy-theoretic techniques to spaces of metrics of positive scalar curvature. Secondly, we relate Hitchin’s secondary index to several other index-theoretical results, such as the Atiyah–Singer family index theorem, the additivity theorem for indices on noncompact manifolds and the spectral flow index theorem. Finally, we use the results and tools developed recently in the study of moduli spaces of manifolds and cobordism categories. The key new ingredient we use in this paper is the high-dimensional analogue of the Madsen–Weiss theorem, proven by Galatius and the third named author.

Semi-simplicial spaces

The generalized Miller‐Morita‐Mumford classes (MMM classes) of a smooth oriented manifold bundle are defined as the image of the characteristic classes of the vertical tangent bundle under the Gysin homomorphism. We show that if the dimension of the manifold is even, then all MMM‐classes in rational cohomology are nonzero for some bundle. In odd dimensions, this is also true with one exception: the MMM‐class associated with the Hirzebruch L‐class is always zero. Moreover, we show that polynomials in the MMM‐classes are also nonzero. We also show a similar result for holomorphic fibre bundles and for unoriented bundles. 55R40