Let $S$ be a surface of genus $2$, and let $S'$ be a surface of genus $3$.
What is an injective homomorphism of $\pi_1(S')$ into $\pi_1(S)$?
2026-03-28 17:04:56.1774717496
Injective homomorphism of $\pi_1$ of genus $3$ surface to $\pi_1$ of genus $2$ surface.
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The two to one covering $p:S' \to S$ defined in the answer to your previous question induces a homomoprhism $$ p_* : \pi_1(S') \to \pi_1(S).$$
This is injective: see here