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Johannes Ebert

Publications and source records attributed to Johannes Ebert.

25 records · Page 2Linked to original sources
arXiv2010

A vanishing theorem for characteristic classes of odd-dimensional manifold bundles

We show how the Atiyah-Singer family index theorem for both, usual and self-adjoint elliptic operators fits naturally into the framework of the Madsen-Tillmann-Weiss spectra. Our main theorem concerns bundles of odd-dimensional manifolds. Using completely functional-analytic methods, we show that for any smooth proper oriented fibre bundle $E \to X$ with odd-dimensional fibres, the family index $\ind (B) \in K^1 (X)$ of the odd signature operator is trivial. The Atiyah-Singer theorem allows us to draw a topological conclusion: the generalized Madsen-Tillmann-Weiss map $α: B \Diff^+ (M^{2m-1}) \to \loopinf \MTSO(2m-1)$ kills the Hirzebruch $\cL$-class in rational cohomology. If $m=2$, this means that $α$ induces the zero map in rational cohomology. In particular, the three-dimensional analogue of the Madsen-Weiss theorem is wrong. For 3-manifolds $M$, we also prove the triviality of $α: B \Diff^+ (M) \to \MTSO (3)$ in mod $p$ cohomology in many cases. We show an appropriate version of these results for manifold bundles with boundary.

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arXiv2010

Algebraic independence of generalized Morita-Miller-Mumford classes

The generalized Morita-Miller-Mumford 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 $\cL$-class is always zero. We also show a similar result for holomorphic fibre bundles.

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arXiv2009

The homotopy type of a topological stack

The notion of the \emph{homotopy type} of a topological stack has been around in the literature for some time. The basic idea is that an atlas $X \to \mathfrak{X}$ of a stack determines a topological groupoid $\mathbb{X}$ with object space $X$. The homotopy type of $\mathfrak{X}$ should be the classifying space $B \mathbb{X}$. The choice of an atlas is not part of the data of a stack and hence it is not immediately clear why this construction of a homotopy type is well-defined, let alone functorial. The purpose of this note is to give an elementary construction of such a homotopy-type functor.

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arXiv2009

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.

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arXiv2008

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

We study the singular homology (with field coefficients) of the moduli stack of stable n-pointed complex curves of genus g (the Deligne-Mumford compactification). Each of its irreducible boundary components determines via the Pontrjagin-Thom construction a map to a certain infinite loop space whose homology is well understood. We show that these maps are surjective on homology in a range of degrees proportional to the genus. This implies the existence of many new torsion classes in the homology of the moduli stack.

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arXiv2007

The low-dimensional homotopy of the stable mapping class group

Due to the deep work of Tillmann, Madsen, Weiss and Galatius, the cohomology of the stable mapping class group $\gaminf$ is known with rational or finite field coefficients. Little is known about the integral cohomology. In this paper, we study the first four cohomology groups. Also, we compute the first few steps of the Postnikov tower of $B \gaminf^+$, the Quillen plus construction applied to $B \gaminf$. Our method relies on the Madsen-Weiss theorem, a few known computations of stable homotopy groups of spheres and projective spaces and on a certain action of the binary icosahedral group on a surface. Using the latter, we can also describe an explicit geometric generator of the third homotopy group $π_3 (B \gaminf)$.

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