The simple loop conjecture (Q760946)

The following is the main result of this paper. Theorem. If \(f: S\to T\) is a map of closed connected surfaces such that \(f_*: \pi_ 1(S)\to \pi_ 1(T)\) is not injective, then there exists a noncontractible simple closed curve \(\alpha\) \(\subset S\) such that \(f| \alpha\) is homotopically trivial. A proof is given of the following known but unpublished result. Theorem. If \(f: S\to T\) is a map of bounded connected compact surfaces such that \(f_*: \pi_ 1(S)\to \pi_ 1(T)\) is not injective, then there exists an essential simple arc \(\alpha\) \(\subset S\) and a map g homotopic to f such that g(\(\alpha)\) is a boundary parallel arc.