Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Coarse Lusternik-Schnirelmann Category of Groups

Aditya De Saha

TL;DR

This work introduces a coarse analogue of Lusternik-Schnirelmann category, $c\text{-}cat$, defined in the coarse homotopy category to study large-scale topological properties of spaces and groups. It proves $c\text{-}cat$ is a coarse homotopy invariant and couples it to existing invariants via two key bounds: $p\text{-}cat(\Gamma) \leq c\text{-}cat(\Gamma)$ for geometrically finite groups, and $c\text{-}cat(\Gamma) \leq \mathrm{asdim}(\Gamma)$ for bicombable $1$-ended groups semistable at $\infty$, establishing a coarse upper bound by asymptotic dimension in substantial cases. The core result shows $\mathrm{c}\text{-}cat(X) \leq \mathrm{asdim}(X)$ for bicombable proper geodesic spaces that are coarsely semistable at infinity, achieved via dispersed-set coverings and an asdim-based construction of coarsely categorical covers. These results deepen the interplay between large-scale topology and coarse geometry, providing new tools for understanding the large-scale structure of groups and spaces with potential implications for geometric group theory and coarse invariants.

Abstract

We introduce a coarse analog of the classical Lusternik-Schnirelmann category which we denote by $\text{c-cat}$, defined for metric spaces in the coarse homotopy category. This provides a new tool for studying large-scale topological properties of groups and spaces. We establish that $\text{c-cat}$ is a coarse homotopy invariant and prove a lower-bound $\text{p-cat}(Γ)\leq \text{c-cat}(Γ)$ for geometrically finite groups $Γ$, where $\text{p-cat}$ denotes the proper LS-category introduced in 1992 by Ayala and co-authors. We also prove an upper bound $\text{c-cat}(Γ) \leq \text{asdim}(Γ)$ for bicombable 1-ended groups which are semistable at $\infty$.

On the Coarse Lusternik-Schnirelmann Category of Groups

TL;DR

This work introduces a coarse analogue of Lusternik-Schnirelmann category, , defined in the coarse homotopy category to study large-scale topological properties of spaces and groups. It proves is a coarse homotopy invariant and couples it to existing invariants via two key bounds: for geometrically finite groups, and for bicombable -ended groups semistable at , establishing a coarse upper bound by asymptotic dimension in substantial cases. The core result shows for bicombable proper geodesic spaces that are coarsely semistable at infinity, achieved via dispersed-set coverings and an asdim-based construction of coarsely categorical covers. These results deepen the interplay between large-scale topology and coarse geometry, providing new tools for understanding the large-scale structure of groups and spaces with potential implications for geometric group theory and coarse invariants.

Abstract

We introduce a coarse analog of the classical Lusternik-Schnirelmann category which we denote by , defined for metric spaces in the coarse homotopy category. This provides a new tool for studying large-scale topological properties of groups and spaces. We establish that is a coarse homotopy invariant and prove a lower-bound for geometrically finite groups , where denotes the proper LS-category introduced in 1992 by Ayala and co-authors. We also prove an upper bound for bicombable 1-ended groups which are semistable at .

Paper Structure

This paper contains 11 sections, 19 theorems, 39 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Coarse Homotopy Category
  3. Coarse Homotopy
  4. Semistability at $\infty$
  5. Coarse Lusternik-Schnirelmann Category
  6. Coarse LS-category vs proper LS-category
  7. Combable Spaces
  8. Asymptotic dimension and $\text{c-cat}$
  9. Dispersed sets and families
  10. Asymptotic Dimension
  11. Acknowledgements

Key Result

Theorem 1.1

For a geometrically finite group $\Gamma$ we have the following inequality

Figures (3)

  • Figure 1: A visual description of the map $\rho : \mathbb{N} \to \mathbb{N}$
  • Figure 2: Illustration of lemma \ref{['lem:technical']}
  • Figure 3: Illustration of $C_{k+1}^{i}$. The shaded blobs are the elements of $C_{k+1}^{i}$.

Theorems & Definitions (56)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Example 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Example 2.10
  • ...and 46 more