0

Let $p\colon \tilde{X} \to X$ be a covering map, and $x_0 \in X$. Suppose that there is a section, i.e. a map $s\colon X \to X$ is a map satisfying $p\circ s = \operatorname{id}_X$. Show that $p_*\colon \pi_1(\tilde{X}, s(x_0)) \to \pi_1(X, x_0)$ is an isomorphism.

I have proved that $p_*$ is a group homomorphism. Furthermore, $p\circ s = \operatorname{id}_X$ gives $p_*\circ s_* = \operatorname{id}_{\pi_1(X, x_0)}$, hence $p_*$ is surjective, but I can't show that $p_*$ is injective.

Sayan Dutta
  • 10,345

1 Answers1

1

It is true because a covering projection which has a section is a homeomorphism. See If a covering map has a section, is it a $1$-fold cover?

Alternatively you can use the well-known fact that $p_*$ is injective for each covering projection. You can find it in any textbook dealing with covering projections.

Paul Frost
  • 87,968