Let $E\to B$ be a fiber bundle (or more genereal, a fibration), and $B'$ be a section.
- Is it true that $\pi_1(B')$ is a subgroup of $\pi_1(E)$?
- Is it true that $\pi_1(B)$ is isomorphic to $\pi_1(B')$?
2026-03-25 06:10:11.1774419011
Fundamental group about fibration
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I claim that $B'$ is a retract of $E$. To see this, let $r: E \to B'$ be the composition $E \to B \to B'$, where the last map is the section map. What does this tell us on the level of fundamental groups?
For the second question, note that the restriction of $E \to B$ to $B'$ is a homeomorphism, as it has an inverse.
EDIT: None of this uses the fact that $E \to B$ is a fiber bundle.