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 the image of 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 be a cover, with and . The homeomorphism

is injective.

Furthermore, if , then is conjugate to .

Conversely, if is a subgroup that is conjugate to , then there exists such that

Proof

todo