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Johannes Ebert

Publications and source records attributed to Johannes Ebert.

At least 19 recordsLinked to original sources
arXiv2025

Cancellation properties for exotic $4$-dimensional positive scalar curvature metrics

Ruberman constructed families $\{g_n\vert n \in \mathbb{N}\} \subset \mathcal{R}^+ (M)$ of metrics of positive scalar curvature on certain $4$-manifolds which are concordant but lie in different path components of $\mathcal{R}^+ (M)$. We prove a cancellation result along the following lines. For each closed manifold $N$, there is a map $\nu_N: \mathcal{R}^+ (M) \to \mathcal{R}^+ (M \times N)$, well-defined up to homotopy, that takes the product with $N$. We prove that when $N$ has positive dimension $\nu_N$ takes all metrics of Ruberman's family to the same path component. This is trivial when $N$ has a psc metric and follows from pseudoisotopy theory when $\dim (N) \geq 3$. Our proof is cobordism theoretic in nature and also applies to $\dim(N) =1,2$. The proof relies on rigidity properties for the action of the diffeomorphism group on $\mathcal{R}^+(L)$ for high-dimensional $N$ and a calculation of $\pi_1(\mathrm{MTSO(4)})$ that we also carry out. Recently, Auckly and Ruberman exhibited examples of elements in higher homotopy groups of $\mathcal{R}^+(M^4)$ for certain $M$. Using the same method, we also prove that these elements lie in the kernel of the induced map $(\nu_N)_*$ on rational homotopy.

math.AT↗
arXiv2024

Tautological classes and higher signatures

For a bundle of oriented closed smooth $n$-manifolds $\pi: E \to X$, the tautological class $\kappa_{\mathcal{L}_k} (E) \in H^{4k-n}(X;\mathbb{Q})$ is defined by fibre integration of the Hirzebruch class $\mathcal{L}_k (T_v E)$ of the vertical tangent bundle. More generally, given a discrete group $G$, a class $u \in H^p(B G;\mathbb{Q})$ and a map $f:E \to B G$, one has tautological classes $\kappa_{\mathcal{L}_k ,u}(E,f) \in H^{4k+p-n}(X;\mathbb{Q})$ associated to the Novikov higher signatures. For odd $n$, it is well-known that $\kappa_{\mathcal{L}_k}(E)=0$ for all bundles with $n$-dimensional fibres. The aim of this note is to show that the question whether more generally $\kappa_{\mathcal{L}_k,u}(E,f)=0$ (for odd $n$) depends sensitively on the group $G$ and the class $u$. For example, given a nonzero cohomology class $u \in H^2 (B \pi_1 (\Sigma_g);\mathbb{Q})$ of a surface group, we show that always $\kappa_{\mathcal{L}_k,u}(E,f)=0$ if $g \geq 2$, whereas sometimes $\kappa_{\mathcal{L}_k,u}(E,f)\neq 0$ if $g=1$. The vanishing theorem is obtained by a generalization of the index-theoretic proof that $\kappa_{\mathcal{L}_k}(E)=0$, while the nontriviality theorem follows with little effort from the work of Galatius and Randal-Williams on diffeomorphism groups of even-dimensional manifolds.

math.GT↗
arXiv2023

Some rational homology computations for diffeomorphisms of odd-dimensional manifolds

We calculate the rational cohomology of the classifying space of the diffeomorphism group of the manifolds $U_{g,1}^n:= \#^g(S^n \times S^{n+1})\setminus \mathrm{int}{D^{2n+1}}$, for large $g$ and $n$, up to approximately degree $n$. The answer is that it is a free graded commutative algebra on an appropriate set of Miller--Morita--Mumford classes. Our proof goes through the classical three-step procedure: (a) compute the cohomology of the homotopy automorphisms, (b) use surgery to compare this to block diffeomorphisms, (c) use pseudoisotopy theory and algebraic $K$-theory to get at actual diffeomorphism groups.

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arXiv2022

On the homotopy type of the space of metrics of positive scalar curvature

Let $M^d$ be a simply connected spin manifold of dimension $d \geq 5$ admitting Riemannian metrics of positive scalar curvature. Denote by $\mathcal{R}^+(M^d)$ the space of such metrics on $M^d$. We show that $\mathcal{R}^+(M^d)$ is homotopy equivalent to $\mathcal{R}^+(S^d)$, where $S^d$ denotes the $d$-dimensional sphere with standard smooth structure. We also show a similar result for simply connected non-spin manifolds $M^d$ with $d\geq 5$ and $d\neq 8$. In this case let $W^d$ be the total space of the non-trivial $S^{d-2}$-bundle with structure group $SO(d-1)$ over $S^2$. Then $\mathcal{R}^+(M^d)$ is homotopy equivalent to $\mathcal{R}^+(W^d)$.

math.DG↗
arXiv2021

Diffeomorphisms of odd-dimensional discs, glued into a manifold

For a compact $(2n+1)$-dimensional smooth manifold, let $μ_M : B Diff_\partial (D^{2n+1}) \to B Diff (M)$ be the map that is defined by extending diffeomorphisms on an embedded disc by the identity. By a classical result of Farrell and Hsiang, the rational homotopy groups and the rational homology of $ B Diff_\partial (D^{2n+1})$ are known in the concordance stable range. We prove two results on the behaviour of the map $μ_M$ in the concordance stable range. Firstly, it is \emph{injective} on rational homotopy groups, and secondly, it is \emph{trivial} on rational homology, if $M$ contains sufficiently many embedded copies of $S^n\times S^{n+1} \setminus int(D^{2n+1})$. The homotopical statement is probably not new and follows from the theory of smooth torsion invariants. The homological statement relies on work by Botvinnik and Perlmutter on diffeomorphism of odd-dimensional manifolds.

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arXiv2020

On the topology of the space of Ricci-positive metrics

We show that the space $\mathcal{R}^{\mathrm{pRc}}(W_g^{2n})$ of metrics with positive Ricci curvature on the manifold $W^{2n}_g := \sharp^g (S^n \times S^n)$ has nontrivial rational homology if $n \not \equiv 3 \pmod 4$ and $g$ are both sufficiently large. The same argument applies to $\mathcal{R}^{\mathrm{pRc}}(W_g^{2n} \sharp N)$ provided that $N$ is spin and $W_g^{2n} \sharp N$ admits a Ricci positive metric.

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arXiv2019

The positive scalar curvature cobordism category

We prove that many spaces of positive scalar curvature metrics have the homotopy type of infinite loop spaces. Our result in particular applies to the path component of the round metric inside $\mathcal{R}^+ (S^d)$ if $d \geq 6$. To achieve that goal, we study the cobordism category of manifolds with positive scalar curvature. Under suitable connectivity conditions, we can identify the homotopy fibre of the forgetful map from the psc cobordism category to the ordinary cobordism category with a delooping of spaces of psc metrics. This uses a version of Quillen's Theorem B and instances of the Gromov--Lawson surgery theorem. We extend some of the surgery arguments by Galatius and the second named author to the psc setting to pass between different connectivity conditions. Segal's theory of $\Gamma$-spaces is then used to construct the claimed infinite loop space structures. The cobordism category viewpoint also illuminates the action of diffeomorphism groups on spaces of psc metrics. We show that under mild hypotheses on the manifold, the action map from the diffeomorphism group to the homotopy automorphisms of the spaces of psc metrics factors through the Madsen--Tillmann spectrum. This implies a strong rigidity theorem for the action map when the manifold has trivial rational Pontrjagin classes. A delooped version of the Atiyah--Singer index theorem proved by the first named author is used to moreover show that the secondary index invariant to real $K$-theory is an infinite loop map. These ideas also give a new proof of the main result of our previous work with Botvinnik.

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arXiv2019

Index theory in spaces of manifolds

We formulate and prove a generalization of the Atiyah-Singer family index theorem in the context of the theory of spaces of manifolds à la Madsen, Tillmann, Weiss, Galatius and Randal-Williams. Our results are for Dirac-type operators linear over arbitrary $C^*$-algebras.

math.AT↗
arXiv2018

The Gromov-Lawson-Chernysh surgery theorem

In this article, we give a complete and self--contained account of Chernysh's strengthening of the Gromov--Lawson surgery theorem for metrics of positive scalar curvature. No claim of originality is made.

math.DG↗
arXiv2018

Semi-simplicial spaces

This is an exposition of homotopical results on the geometric realization of semi-simplicial spaces. We then use these to derive basic foundational results about classifying spaces of topological categories, possibly without units. The topics considered include: fibrancy conditions on topological categories; the effect on classifying spaces of freely adjoining units; approximate notions of units; Quillen's Theorems A and B for non-unital topological categories; the effect on classifying spaces of changing the topology on the space of objects; the Group-Completion Theorem.

math.AT↗
arXiv2017

Infinite loop spaces and positive scalar curvature in the presence of a fundamental group

This is a continuation of our previous work with Botvinnik on the nontriviality of the secondary index invariant on spaces of metrics of positive scalar curvature, in which we take the fundamental group of the manifolds into account. We show that the secondary index invariant associated to the vanishing of the Rosenberg index can be highly nontrivial, for positive scalar curvature Spin manifolds with torsionfree fundamental groups which satisfy the Baum--Connes conjecture. For example, we produce a compact Spin 6-manifold such that its space of positive scalar curvature metrics has each rational homotopy group infinite dimensional. At a more technical level, we introduce the notion of "stable metrics" and prove a basic existence theorem for them, which generalises the Gromov--Lawson surgery technique, and we also give a method for rounding corners of manifold with positive scalar curvature metrics.

math.AT↗
arXiv2017

Construction of Non-asymptotic Confidence Sets in 2-Wasserstein Space

In this paper, we consider a probabilistic setting where the probability measures are considered to be random objects. We propose a procedure of construction non-asymptotic confidence sets for empirical barycenters in 2-Wasserstein space and develop the idea further to construction of a non-parametric two-sample test that is then applied to the detection of structural breaks in data with complex geometry. Both procedures mainly rely on the idea of multiplier bootstrap (Spokoiny and Zhilova (2015), Chernozhukov et al. (2014)). The main focus lies on probability measures that have commuting covariance matrices and belong to the same scatter-location family: we proof the validity of a bootstrap procedure that allows to compute confidence sets and critical values for a Wasserstein-based two-sample test.

math.ST↗
arXiv2017

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 $2n \geq 6$, 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 $2n+1 \geq 7$, 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.

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arXiv2016

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.

math.KT↗
arXiv2015

Stable cohomology of the universal Picard varieties and the extended mapping class group

We study the moduli spaces which classify smooth surfaces along with a complex line bundle. There are homological stability and Madsen--Weiss type results for these spaces (mostly due to Cohen and Madsen), and we discuss the cohomological calculations which may be deduced from them. We then relate these spaces to (a generalisation of) Kawazumi's extended mapping class groups, and hence deduce cohomological information about these. Finally, we relate these results to complex algebraic geometry. We construct a holomorphic stack classifying families of Riemann surfaces equipped with a fibrewise holomorphic line bundle, which is a gerbe over the universal Picard variety, and compute its holomorphic Picard group.

math.AT↗
arXiv2015

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.

math.AT↗
arXiv2014

Torelli spaces of high-dimensional manifolds

The Torelli group of a manifold is the group of all diffeomorphisms which act as the identity on the homology of the manifold. In this paper, we calculate the invariant part (invariant under the action of the automorphisms of the homology) of the cohomology of the classifying space of the Torelli group of certain high-dimensional, highly connected manifolds, with rational coefficients and in a certain range of degrees. This is based on Galatius--Randal-Williams' work on the diffeomorphism groups of these manifolds, Borel's classical results on arithmetic groups, and methods from surgery theory and pseudoisotopy theory. As a corollary, we find that all Miller--Morita--Mumford characteristic classes are nontrivial in the cohomology of the classifying space of the Torelli group, except for those associated with the Hirzebruch class, whose vanishing is forced by the family index theorem.

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