Ends and end cohomology

Abstract

Ends and end cohomology are powerful invariants for the study of noncompact spaces. We present a self-contained exposition of the topological theory of ends and prove novel extensions including the existence of an exhaustion of a proper map. We define reduced end cohomology as the relative end cohomology of a ray-based space. We use those results to prove a version of a theorem of King that computes the reduced end cohomology of an end sum of two manifolds. We include a complete proof of Freudenthal’s fundamental theorem on the number of ends of a topological group, and we use our results on dimension-zero end cohomology to prove—without using transfinite induction—a theorem of Nöbeling on freeness of certain modules of continuous functions.

Publisher

Elsevier

Publication Date

9-1-2025

Publication Title

Expositiones Mathematicae

Department

Mathematics

Document Type

Article

DOI

https://doi.org/10.1016/j.exmath.2025.125692

Keywords

Ends, Naturality, End cohomology, Ray-based space, End sum, Profinite

Language

English

Format

text

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