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Dec 10, 2021 at 9:00 vote accept G. Blaickner
Dec 9, 2021 at 20:33 answer added Moishe Kohan timeline score: 5
Dec 9, 2021 at 14:21 comment added Moishe Kohan Of course not, take the exterior of any nontrivial knot in $S^3$. What is true is that either your manifold admits a nontrivial boundary connected sum decomposition, or it is the solid torus connected sum with a closed manifold, or it has incompressible boundary.
Dec 9, 2021 at 10:04 history edited G. Blaickner CC BY-SA 4.0
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Dec 9, 2021 at 9:56 history asked G. Blaickner CC BY-SA 4.0