3-manifold
Simple loop conjecture

If f: S → T {\displaystyle f\colon S\rightarrow T} is a map of closed connected surfaces such that f ⋆: π 1 (S) → π 1 (T) {\displaystyle f_{\star }\colon \pi _{1}(S)\rightarrow \pi _{1}(T)} is not injective, then there exists a non-contractible simple closedcurve α ⊂ S {\displaystyle \alpha \subset S} such that f | a {\displaystyle f|_{a}} is homotopically trivial. This conjecture was proven by David Gabai.
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