Ends and end cohomology

TL;DR

This paper develops a comprehensive topological framework for ends and end cohomology, extending existing theories and applying them to compute invariants of noncompact spaces and groups.

Contribution

It introduces novel extensions of end cohomology, including exhaustion of proper maps and reduced end cohomology, and applies these to classical theorems in topology.

Findings

Proves existence of exhaustion of proper maps.

Computes reduced end cohomology of end sums of manifolds.

Provides a new proof of Freudenthal's theorem on the number of ends.

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 N\"obeling on freeness of certain modules of continuous functions.