Johannes Ebert
University of Bonn
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Featured researches published by Johannes Ebert.
Inventiones Mathematicae | 2017
Boris Botvinnik; Johannes Ebert; Oscar Randal-Williams
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
Algebraic & Geometric Topology | 2011
Johannes Ebert
Algebraic & Geometric Topology | 2014
Johannes Ebert; Oscar Randal-Williams
2n \ge 6
Topology and its Applications | 2008
Johannes Ebert; Oscar Randal-Williams
Algebraic & Geometric Topology | 2008
Johannes Ebert; Jeffrey Giansiracusa
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
arXiv: Algebraic Topology | 2012
Johannes Ebert; Oscar Randal-Williams
Mathematische Annalen | 2011
Johannes Ebert; Jeffrey Giansiracusa
2n+1 \ge 7
arXiv: Algebraic Topology | 2009
Johannes Ebert
Archive | 2006
Johannes Ebert
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.
arXiv: Algebraic Topology | 2017
Johannes Ebert; Oscar Randal-Williams
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