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Thorben Kastenholz

Publications and source records attributed to Thorben Kastenholz.

10 recordsLinked to original sources
arXiv2025

Non vanishing of the fourth bounded cohomology of free groups and codimension 2 subspaces

In this note we prove that the fouth bounded cohomology of non-abelian free groups with trivial real coefficients is non-zero. In order to prove this, we establish a splitting argument whose simplest form is as follows: Let $M$ denote an $n$-manifold of non-zero simplicial volume and $S$ a codimension two submanifold of $M$, then one can conclude that the $n$-th bounded cohomology of the fundamental group of $M \setminus S$ is non-zero. While in this note this approach is only used for degree $4$. There is no reason to expect that this approach and its generalizations is not suitable to prove the non-vanishing of higher degrees or the bounded cohomology of different groups as well.

math.GR↗
arXiv2025

Simplicial volume of open books in dimension 4

In this short note we adapt a proof by Bucher and Neofytidis to prove that the simplicial volume of 4-manifolds admitting an open book decomposition vanishes. In particular this shows that Quinns signature invariant, which detects the existence of an open book decomposition in dimensions above 4, is insufficient to characterize open books in dimension 4, even if one allows arbitrary stabilizations via connected sums.

math.GT↗
arXiv2024

Simplicial volume of manifolds fibering with connected structure group

In this note we investigate the simplicial volume of fiber bundles with connected structure group. We are able to show that if the structure group is either compact or a Lie group, or if the fiber is aspherical that the simplicial volume of the total space agrees with the simplicial volume of the trivial bundle.

math.AT↗
arXiv2023

Simplicial bounded cohomology and stability

We introduce a set of combinatorial techniques for studying the simplicial bounded cohomology of semi-simplicial sets, simplicial complexes and posets. We apply these methods to prove several new bounded acyclicity results for semi-simplicial sets appearing in the homological stability literature. Our strategy is to recast classical arguments (due to Bestvina, Maazen, van der Kallen, Vogtmann, Charney and, recently, Galatius--Randal-Williams) in the setting of bounded cohomology using uniformly bounded refinements of well-known simplicial tools. Combined with ideas developed by Monod and De la Cruz Mengual--Hartnick, we deduce slope-$1/2$ stability results for the bounded cohomology of two large classes of linear groups: general linear groups over any ring with finite Bass stable rank and certain automorphism groups of quadratic modules over the integers or any field of characteristic zero. We expect that many other results in the literature on homological stability admit bounded cohomological analogues by applying the blueprint provided in this work.

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arXiv2023

Symplectic Groups, Mapping Class Groups and the Stability of Bounded Cohomology

Mapping class groups satisfy cohomological stability. In this note we show how results by Bestvina and Fujiwara imply that the bounded cohomology does not stabilize, additionally we show that stabily polynomials in the Mumford-Morita-Miller classes are unbounded i.e. their norm tends to infinity as one increases the genus. While the bounded cohomology of the symplectic group does stabilize, we show that it does not stabilize via isometries in degree $2$. In order to establish this we calculate the norm of the signature class in $\mathrm{Sp}_{2h}(\mathbb{R})$ and estimate the norm of the integral signature class.

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arXiv2023

The minimal genus problem for right angled Artin groups

We investigate the minimal genus problem for the second homology of a right angled Artin group (RAAG). Firstly, we present a lower bound for the minimal genus of a second homology class, equal to half the rank of the corresponding cap product matrix. We show that for complete graphs, trees, and complete bipartite graphs, this bound is an equality, and furthermore in these cases the minimal genus can always be realised by a disjoint union of tori. Additionally, we give a full characterisation of classes that are representable by a single torus. However, the minimal genus of a second homology class of a RAAG is not always realised by a disjoint union of tori as an example we construct in the pentagon shows.

math.GT↗
arXiv2022

Simplicial volume and essentiality of manifolds fibered over spheres

We study the question when a manifold that fibers over a sphere can be rationally essential, or even have positive simplicial volume. More concretely, we show that mapping tori of manifolds (whose fundamental groups can be quite arbitrary) of odd dimension at least 7 with non-zero simplicial volume are very common. This contrasts the case of fiber bundles over a sphere of dimension d > 1: we prove that their total spaces are rationally inessential if d is at least 3, and always have simplicial volume 0. Using a result by Dranishnikov, we also deduce a surprising property of macroscopic dimension, and we give two applications to positive scalar curvature and characteristic classes, respectively.

math.GT↗
arXiv2020

Homological Stability for Spaces of Subsurfaces with Tangential Structure

Given a manifold with boundary, one can consider the space of subsurfaces of this manifold meeting the boundary in a prescribed fashion. It is known that these spaces of subsurfaces satisfy homological stability if the manifold has at least dimension five and is simply-connected. We introduce a notion of tangential structure for subsurfaces and give a general criterion for when the space of subsurfaces with tangential structure satisfies homological stability provided that the manifold is simply-connected and has dimension $n\geq 5$. Examples of tangential structures such that the spaces of subsurface with that tangential structure satisfy homological stability are framings or spin structures of their tangent bundle, or $k$-frames of the normal bundle provided that $k\leq n-2$. Furthermore we introduce spaces of pointedly embedded subsurfaces and construct stabilization maps, as well as prove homological stability for these. This is used to prove homological stability for spaces of symplectic subsurfaces.

math.AT↗
arXiv2020

The Minimal Genus of Homology Classes in a Finite 2-Complex

We study surface representatives of homology classes of finite complexes which minimize certain complexity measures, including its genus and Euler characteristic. Our main result is that up to surgery at nullhomotopic curves minimizers are homotopic to cellwise coverings to the 2-skeleton. From this we conclude that the minimizing problem is in general algorithmically undecidable, but can be solved for 2-dimensional CAT(-1)-complexes.

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