Methods of constructing spaces with non-trivial self covers

TL;DR

This survey catalogs methods for constructing spaces with nontrivial self-covers, organizing them into five broad families: products and bundles (including Seifert-fibred and fibered constructions), categorical (e.g., Delgado–Timm and Tinsley) approaches, Cantor-set–controlled constructions, graphs of spaces (GBS frameworks), and inverse-limit/solenoid-based techniques. It highlights how these methods interact with group-theoretic notions like non-cohopficity and finite-index clones, and it emphasizes obstructions from fixed-point properties and semi-local simply connectedness. Central examples include tori, Klein bottles, solenoids, Cantor-based continua with $G$-regular self-covers for finite groups $G$, and various 3- and 4-manifold constructions whose covering spaces realize geometric manifestations of algebraic non-cohopfian behavior. The paper also discusses the Hawaiian Earring as a cautionary counterexample and poses open questions about classification, realizability, and the limits of current constructions, inviting further examples for future work.

Abstract

We survey techniques for constructing spaces with non-trivial self covers. These process include methods for building low and high dimension continua which non-trivially self. We also discuss several related group theoretic and topological concerns. Statements of a number of open problems related to self covering phenomena are included.

Figures (18)

• Figure 1: The necklace of circles $\mathcal{N}^{(1)}_{C,S^1}$ and its $\mathbb{Z}_3$-regular cover $f_3:\mathcal{N}^{(1)}_{C,S^1}\rightarrow \mathcal{N}^{(1)}_{C,S^1}$. The vertical line segments in the picture corresponding to the rational points of the Cantor set are not part of the necklace. They are included only for reference. The lifts $f^{-1}_3(0=2)$ in the domain copy are indicated by the longest vertical line segments in the domain copy of $\mathcal{N}^{(1)}_{C,S^1}$. These line segments delineate closures of fundamental domains of the covering map. Note that on each of these fundamental domains $f_3$ is non-contractive. In fact, it is expansive on the part of the domain copy of $\mathcal{N}^{(1)}_{C,S^1}$ between $0=2$ and $1$ in the picture and maps the ellipse sitting between $1$ and $2=0$ homeomorphically onto itself.
• Figure 2: The 1-dimensional continuum $\mathcal{N}^{(2)}_{C,S^1}$ with $\mathbb{Z}_m$-regular self covers for all $m\in \mathbb{Z}.$ The small vertical line segments corresponding to the rational points of the Cantor set, the one at $\frac{5}{4}$, and the one at $\frac{7}{4}$ are not part of $\mathcal{N}^{(2)}_{C,S^1}$. They are included only to indicate the location of the associated rational points. Should the reader be inclined to include them, doing so produces another continuum with $\mathbb{Z}_m$-regular self covers for every $m \in \mathbb{N}$. In this case the control set is $C\times I$.
• Figure 3: The continuum $\mathcal{N}^{(3)}_{C\times [-1,1],S^1}.$ It has a $\mathbb{Z}_m$-regular self covers for each $m\in\mathbb{Z}$. The $x$-coordinates at the points $\frac{i}{27}$ have been removed to reduce clutter in the picture. The control set in this case is the product $C\times [-1,1]$. The direction of the gluing on $0\times [-1,1]$ and $2\times [-1,1]$ are indicated by the arrows in the picture.
• Figure 4: Cantor's Pearl Necklace $\mathcal{N}^{(2)}_{C,S^2}$. The vertical line segments in the picture corresponding to the rational points of the Cantor set are included only for reference and are not part of the Pearl Necklace. However, should the reader wish to include them, this does produces another necklace with $\mathbb{Z}_m$-regular self covers for every $m \in \mathbb{Z}$.
• Figure 5: The circle-like Knaster Cup-cap continuum. The dashed blue in the picture indicate the closures of fundamental domains of a $\mathbb{Z}_3$-regular self cover. The arrows on the segments $0\times[-1,1]$ and $2\times [-1,1]$ show the gluing map $(0,t)\mapsto (2,t)$ used to form the circle-like Cup-cap continuum.

Theorems & Definitions (10)

• Example 3.1
• Example 3.2
• Example 3.3
• Example 3.4
• Theorem 1
• Theorem 2
• Example 4.1
• Example 7.1
• Definition 1
• Definition 2