(Theorem) Classification of covering maps

Overview

The classification question for all path-connected coverings for a fixed path-connected space $(Y,y_0)$ involves finding all possible triples $(X,x,p),$ where $(X,x)$ is another path-connected space and $p:X\to Y$ is a covering map for which $p(x)=y_0$, up to isomorphism.

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Isomorphisms of covering spaces

Isomorphism of covering spaces

Given a base space $X$, two covering spaces $p:Y\to X$ and $p’:Y’\to X$ are isomorphic if there exists a based homeomorphism $h:(Y,y)\to (Y’,y^{\prime })$ such that $p=p^{\prime }\circ h.$ More precisely, the map $h$ preserves the covering space structure by sending $p^{-1}(x)\to (p’{)}^{-1}(x)$ for all $x\in X$.

Hatcher 1.37: Isomorphism of covering spaces is an equivalence relation

Two path-connected coverings $p:Y\to X$ and $p’:Y’\to X$ are isomorphic (i.e., there exists a based homeomorphism $h:(Y,y)\to (Y’,y^{\prime })$ such that $p=p’\circ h$) if and only if the images of their induced homomorphisms $p_{\ast }:{\pi }_1(Y,y)\to {\pi }_1(X,x)p_{\ast }^{\prime }:{\pi }_1(Y,y)\to {\pi }_1(X,x)$ are equivalent: $p_{\ast }([f])=[p\circ f]=[p^{\prime }\circ h\circ f]=p_{\ast }^{\prime }([h\circ f]).$ In this case, the homeomorphism $h$ is unique.

2025-02-11: More general method in both directions to find the homeomorphism

• Uniqueness part of the classification theorem (injective and well-defined?)

Proof from Algebraic Topology.

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Classification theorem statement and proof

Semi-locally simply connected

A space $X$ is semi-locally simply connected if for all $x\in X$, there exists a path-connected neighborhood $U\subseteq X$ of $x$ such that either of the equivalent statements hold:

• (i) The group homomorphism induced by inclusion is $i_{\ast }:{\pi }_1(U,x)\to {\pi }_1(X,x)$ is trivial, i.e., the class of every loop in $U$ is mapped to the class of the constant map at $x$;
• (ii) Every loop in $U$ is nullhomotopic, i.e., can be contracted to a single point in $X$.

Hatcher 1.38: Classification of covering spaces

Let $(Y,y_0)$ be a based space that is path-connected, locally path-connected, and semi-locally simply connected. Then there is a bijection between the set $\{(X,x,p):X\text{path-connected},x\in X,p:X\to Y\text{a covering s.t.}p(x)=y_0\}/\sim ,$ where $\sim$ denotes the isomorphism classes of $(X,x,p)$ (hence $p(x)=y_0$ is required up to the equivalence relation $\sim$), and the set of subgroups of ${\pi }_1(Y,y_0)$, i.e., $\{H\le {\pi }_1(Y,y_0):H\text{is a subgroup}\}.$

• Wdym universal covering exists (simply-connected) coveirng and is unique up to isomorphism?

and $p:(Y,y_0)\to (X,x_0)$ be a based covering map. Then given some “mild assumptions” on the base space $X$ (see top of page), we have:

• H exists iff the images are the same
• Any subgroup of fundamental group arises as the image

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Examples

Fundamental group of the circle

Recall that any subgroup $H\le d\mathbb{Z}$ is of the form $d\ge 1$ for $d\in \mathbb{N}$. Consider the map $p:\mathbb{R}\to S^1$ defined by

$$p(t)=(\cos 2\pi t,\sin 2\pi t).$$

Any two loops in $\mathbb{R}$ are homotopic rel $\partial I$, so ${\pi }_1(\mathbb{R},0)=\{e\}$ is trivial and thus the image of the homomorphism induced by $p$ is $\text{Im}(p_{\ast })=\{e\}=0\mathbb{Z}\le \mathbb{Z}.$

concept-question what

Now let $p_d:S^1\to S^1$ be the map defined by “wrapping around the circle $d$ times”:

$$p_d((\cos 2\pi t,\sin 2\pi t))=(\cos 2\pi dt,\sin 2\pi dt).$$

By the same reasoning, the image of its induced homomorphism is

$\text{Im}((p_d{)}_{\ast })=d\mathbb{Z}\le \mathbb{Z}.$

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Notes

2025-02-22

• Any two path-connected covering spaces of a base space $X$ can be classified by the following fact: there is a (based) homeomorphism between them in and only if the images of their fundamental groups under their induced homomorphisms—that is, as subgroups of the fundamental group of $X$—are equivalent.

2025-02-06

• $G$ from (Theorem) A group acting on a simply connected space is isomorphic to the fundamental group of its orbits exists iff the images are equal
• Any subgroup $H\le {\pi }_1(X,x)$ arises as the image of the homomorphism induced by a based covering map $p:(Y,y)\to (X,x)$.
• In fact this is an iff relation?