A sufficient condition for surfaces in 3-manifolds to have unique prime decompositions

Motto, Michael

doi:10.1090/S0002-9939-1994-1195727-8

A sufficient condition for surfaces in $3$-manifolds to have unique prime decompositions
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Proc. Amer. Math. Soc. 120 (1994), 1275-1280 Request permission

Abstract:

In 1975, Suzuki proved that prime decompositions of closed, connected surfaces in ${S^3}$ are unique up to ambient isotopy if the surface bounds a $3$-manifold whose factors under the prime decomposition all have incompressible boundary. This paper extends this result to surfaces in more general $3$-manifolds, when there is a prime decomposition for which every factor of the surface is incompressible on one side.

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