Homotopy and Homology at Infinity and at the Boundary

2021/10/30 by Mohammed Barhoush, Barhoush, Mohammed
Mathematics · #Advanced Combinatorial Mathematics #Algebraic Topology (math.AT) #Boundary (topology) #Cohomology #Conjecture #FOS: Mathematics #Geometric and Algebraic Topology #Group (periodic table) #Homology (biology) #Homotopy #Homotopy and Cohomology in Algebraic Topology #Homotopy group #Infinity #Mathematical analysis #Mathematics #Physics #Pure mathematics #Singular homology #math.AT

paper · pdf · doi:10.48550/arxiv.2111.00342

arxiv created 2021/10/30 · openalex publication_date 2021/10/30 · arxiv updated 2021/11/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we study the relationship between the homology and homotopy of a space at infinity and at its boundary. Firstly, we prove that if a locally connected, connected, δ-hyperbolic space that is acted upon geometrically by a group has trivial homotopy at infinity then the first Čech homotopy group is trivial. Secondly, we prove that if a hyperbolic group on a finite field has trivial i^th homology at infinity then the boundary of the group has trivial i^th Steenrod homology. This result turns out to be important in addressing an open problem related to Cannon's conjecture.

Citations

• Cut points and canonical splittings of hyperbolic groups
• A Unique Decomposition Theorem for 3-Manifolds
• Compact Submanifolds of 3-Manifolds
• Metric Spaces of Non-Positive Curvature
• Tree amalgamations and quasi-isometries

Related

• Ends, shapes, and boundaries in manifold topology and geometric group theory
• Eilenberg-Steenrod homology and cohomology theories for Čech's closure spaces
• Tameness on the boundary and Ahlfors' measure conjecture
• On the Čech cohomology of Morse boundaries
• Hyperbolic boundaries vs. hyperbolic groups