induced subgroup from cover

Overview

There is a connection between the fundamental group of a covering space and that of the base space.

Moving the point around in the fiber “moves” the image subgroup of the pushforward of the cover through conjugate subgroups.

This means that for a cover $p:E\to X$ the image of ${\pi }_1(E)$ can be seen as a congugacy class of subgroups. Furthermore, it is an invariant of cover (doesn’t change with any given point in the fiber.)

Statement

Let $p:E\to X$ be a cover, with $x\in X$ and $e\in p^{-1}(x)$. The homeomorphism

$p_{\ast }:{\pi }_1(E,e)⟶{\pi }_1(X,x)$

is injective.

Furthermore, if $e^{\prime }\in p^{-1}(x)$, then $p_{\ast }{\pi }_1(E,e^{\prime })$ is conjugate to $p_{\ast }{\pi }_1(E,e)$.

Conversely, if $H\subset {\pi }_1(X,x)$ is a subgroup that is conjugate to $p_{\ast }{\pi }_1(E,e)$, then there exists $e^{\prime }\in E$ such that

$H=p_{\ast }{\pi }_1(E,e^{\prime }).$

Proof

todo