Ends and end cohomology

TL;DR

This work develops a self-contained theory of ends and end cohomology for generalized continua, introducing reduced end cohomology via baserays and establishing a natural splitting of end cohomology for ray-based maps. It proves a King-type formula expressing the end cohomology of an end sum $(M,r)\natural (N,s)$ in terms of the summands, along with a precise count of ends, $|E(S)|=|E(M)|+|E(N)|-1$. The paper provides foundational background on compact exhaustions, end spaces, Freudenthal’s endpoint compactification, baserays, and retracts to baserays, and derives a direct, countable-rank description of $H^{0}_{\mathrm{e}}(X)$ enabling a direct proof of Nöbeling’s theorem. It also analyzes the dependence of reduced end cohomology on ray choice and outlines avenues for extending the theory to trees and diffeomorphism-type questions for end sums, indicating substantial impact for topology of noncompact spaces and manifold construction.

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.

Figures (16)

• Figure 2.2: Connected compactum $K$ in $\mathbb{R}^2$.
• Figure 3.3: Compactum $K_k$ (blue) in three-ended space $X$ (left). Image of $K_k$ (blue) under left translation $L_g$ of $X$ and connected subspace $K_k \cup V_k^2 \cup V_k^3$ (yellow) of $X$.
• Figure 4.1: Graph of a piecewise linear, proper map $f:[0,\infty)\to \mathbb{R}$.
• Figure 4.2: Closed subspace $A$ of $X$ and shuffled efficient compact exhaustions $\left\{L_i\right\}$ of $A$ and $\left\{J_i\right\}$ of $X$.
• Figure 5.1: Three basic open sets for the dictionary order topology on $X=[0,1)\times[0,1]$.

Theorems & Definitions (116)

• Example 2.4
• Lemma 2.5
• proof
• Lemma 2.6
• proof
• Lemma 2.7
• proof
• Lemma 2.8: Monotonicity
• proof
• Theorem 2.9: Existence of efficient compact exhaustion