Searcharxiv⌕ Search

arXiv subjects

Elisa Hartmann

Publications and source records attributed to Elisa Hartmann.

11 recordsLinked to original sources
arXiv2024

Geometric invariants of locally compact groups: the homological perspective

In this paper we develop the homological version of $\Sigma$-theory for locally compact Hausdorff groups, leaving the homotopical version for another paper. Both versions are connected by a Hurewicz-like theorem. They can be thought of as directional versions of type $\mathrm{CP}_m$ and type $\mathrm{C}_m$, respectively. And classical $\Sigma$-theory is recovered if we equip an abstract group with the discrete topology. This paper provides criteria for type $\mathrm{CP}_m$ and homological locally compact $\Sigma^m$. Given a short exact sequence with kernel of type $\mathrm{CP}_m$, we can derive $\Sigma^m$ of the extension on the sphere that vanishes on the kernel from the quotient and likewise. Given a short exact sequence with abelian quotient, $\Sigma$-theory on the extension can tell if the kernel is of type $\mathrm{CP}_m$.

math.AT↗
arXiv2024

Geometric invariants of locally compact groups: the homotopical perspective

We extend the classical theory of homotopical $\Sigma$-sets $\Sigma^n$ developed by Bieri, Neumann, Renz and Strebel for abstract groups, to $\Sigma$-sets $\Sigma_{\mathrm{top}}^n$ for locally compact Hausdorff groups. Given such a group $G$, our $\Sigma_{\mathrm{top}}^n(G)$ are sets of continuous homomorphisms $G \to \mathbb{R}$ ("characters"). They match the classical $\Sigma$-sets $\Sigma^n(G)$ if $G$ is discrete, and refine the homotopical compactness properties $\mathrm C_n$ of Abels and Tiemeyer. Moreover, our theory recovers the definition of $\Sigma_{\mathrm{top}}^1$ and $\Sigma_{\mathrm{top}}^2$ proposed by Kochloukova. Besides presenting various characterizations of $\Sigma_{\mathrm{top}}^n$ (particularly for $n\in \{1,2\}$), we show that characters in $\Sigma_{\mathrm{top}}^n(G)$ are also in $\Sigma_{\mathrm{top}}^n(H)$ if $H\le G$ is a closed cocompact subgroup, and we generalize several classical results. Namely, we prove that the set of nonzero elements of $\Sigma_{\mathrm{top}}^n(G)$ is open, we prove that characters in a group of type $\mathrm C_n$ that do not vanish on the center always lie in $\Sigma_{\mathrm{top}}^n(G)$, and we relate the $\Sigma$-sets of a group with those of its quotients by closed subgroups of type $\mathrm C_n$. Lastly, we describe how $\Sigma_{\mathrm{top}}^n(G)$ governs whether a closed normal subgroup with abelian quotient is of type $\mathrm C_n$, generalizing one of the highlights of the classical theory.

math.GR↗
arXiv2024

Computer-assisted methods in Sigma-theory

We develop an algorithm for recognizing whether a character belongs to $\Sigma^m$. In order to apply it we just need to know that the ambient group is of type $\mathrm{FP}_m$ or of type $\mathrm{F}_2$ and that the word problem is solvable for this group. Then finite data is sufficient proof of membership in $\Sigma^m$, not just for the given character but also for a neighborhood of it.

math.GR↗
arXiv2022

Coarse sheaf cohomology

A certain Grothendieck topology assigned to a metric space gives rise to a sheaf cohomology theory which sees the coarse structure of the space. Already constant coefficients produce interesting cohomology groups. In degree 0 they see the number of ends of the space. In this paper a resolution of the constant sheaf via cochains is developed. It serves to be a valuable tool for computing cohomology. In addition coarse homotopy invariance of coarse cohomology with constant coefficients is established. This property can be used to compute cohomology of Riemannian manifolds. The Higson corona of a proper metric space is shown to reflect sheaves and sheaf cohomology. Thus we can use topological tools on compact Hausdorff spaces in our computations. In particular if the asymptotic dimension of a proper metric space is finite then higher cohomology groups vanish. We compute a few examples. As it turns out finite abelian groups are best suited as coefficients on finitely generated groups.

math.AT↗
arXiv2020

Coarse compactifications of proper metric spaces

This paper studies coarse compactifications and their boundary. We introduce two alternative descriptions to Roe's original definition of coarse compactification. One approach uses bounded functions on $X$ that can be extended to the boundary. They satisfy the Higson property exactly when the compactification is coarse. The other approach defines a relation on subsets of $X$ which tells when two subsets closure meet on the boundary. A set of axioms characterizes when this relation defines a coarse compactification. Such a relation is called large-scale proximity. Based on this foundational work we study examples for coarse compactifications Higson compactification, Freudenthal compactification and Gromov compactification. For each example we characterize the bounded functions which can be extended to the coarse compactification and the corresponding large-scale proximity relation. We provide an alternative proof for the property that the Higson compactification is universal among coarse compactifications. Furthermore the Freudenthal compactification is universal among coarse compactifications with totally disconnected boundary. If $X$ is hyperbolic geodesic proper then there is a closed embedding $\nu(\mathbb R_+)\times \partial X\to \nu(X)$. Its image is a retract of $\nu(X)$ if $X$ is a tree.

math.MG↗
arXiv2019

Coarse Homotopy on metric Spaces and their Corona

This paper discusses properties of the Higson corona by means of a quotient on coarse ultrafilters on a proper metric space. We use this description to show that the corona functor is faithful. This study provides a K\"unneth formula for twisted coarse cohomology. We obtain the Gromov boundary of a hyperbolic proper geodesic metric space as a quotient of its Higson corona.

math.MG↗
arXiv2019

A pullback diagram in the coarse category

This paper studies the asymptotic product of two metric spaces. It is well defined if one of the spaces is visual or if both spaces are geodesic. In this case the asymptotic product is the pullback of a limit diagram in the coarse category. Using this product construction we can define a homotopy theory on coarse metric spaces in a natural way. We prove that all finite colimits exist in the coarse category.

math.MG↗
arXiv2019

Twisted Coefficients on coarse Spaces and their Corona

To a metric space $X$ we associate a compact topological space $\nu' X$ called the corona of $X$. Then a coarse map $f:X\to Y$ between metric spaces is mapped to a continuous map $\nu' f:\nu' X\to \nu' Y$ between coronas. Sheaf cohomology on coarse spaces has been introduced in arXiv:1710.06725. We show the functor $\nu'$ preserves and reflects sheaf cohomology.

math.MG↗
arXiv2017

A twisted Version of controlled K-Theory

There are a number of (co-)homology theories on coarse spaces. Controlled operator K-theory is by far the most popular one of them. Our approach is geometric. We study when does the Roe-algebra of a space restrict to a subspace. Then we show the Roe-algebra is a cosheaf on the coarse topology. A result is a Mayer-Vietoris exact sequence in the presence of a coarse cover. We compute examples as an application.

math.KT↗
arXiv2017

Coarse Cohomology with twisted Coefficients

To a coarse structure we associate a Grothendieck topology which is determined by coarse covers. A coarse map between coarse spaces gives rise to a morphism of Grothendieck topologies. This way we define sheaves and sheaf cohomology on coarse spaces. We obtain that sheaf cohomology is a functor on the coarse category: if two coarse maps are close they induce the same map in cohomology. There is a coarse version of a Mayer-Vietoris sequence and for every inclusion of coarse spaces there is a coarse version of relative cohomology. Cohomology with constant coefficients can be computed using the number of ends of a coarse space.

math.AG↗