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 N\"obeling on freeness of certain modules of continuous functions.
Cite
@article{arxiv.2412.01816,
title = {Ends and end cohomology},
author = {William G. Bass and Jack S. Calcut},
journal= {arXiv preprint arXiv:2412.01816},
year = {2025}
}
Comments
56 pages, 21 figures. Included acknowledgement, replaced Figure 4.2, and corrected a few typos