Locally finite graphs with ends: a topological approach. III. Fundamental group and homology

TL;DR

This paper explores the algebraic topology of locally finite graphs with ends, extending classical finite graph theorems to an infinite setting using topological arcs, circles, fundamental groups, and homology.

Contribution

It provides a combinatorial description of the fundamental group and homology of locally finite graphs with ends, advancing the topological understanding of infinite graphs.

Findings

Describes the fundamental group of locally finite graphs with ends.

Analyzes homology aspects of these graphs.

Extends classical finite graph theorems to infinite graphs.

Abstract

This paper is the last part of a comprehensive survey of a newly emerging field: a topological approach to the study of locally finite graphs that crucially incorporates their ends. Topological arcs and circles, which may pass through ends, assume the role played in finite graphs by paths and cycles. The first two parts of the survey together provide a suitable entry point to this field for new readers; they are available in combined form in arXiv:0912.4213. The topological approach indicated above has made it possible to extend to locally finite graphs many classical theorems of finite graph theory that do not extend verbatim. While the first part of this survey introduces the theory as such and the second part is devoted to those applications, this third part looks at the theory from an algebraic-topological point of view. The results surveyed here include both a combinatorial description of the fundamental group of a locally finite graph with ends and the homology aspects of this space.