Celebratio Mathematica — Freedman

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Works connected to M. Benedict Freedman

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article B. Freedman and M. H. Freedman: “Kneser–Haken finiteness for bounded 3-manifolds locally free groups, and cyclic covers,” Topology 37 : 1 (1998), pp. 133–147. MR 1480882 Zbl 0896.57012

Associated to every compact 3-manifold \( M \) and positive integer \( b \), there is a constant \( c(M,b) = c \). Any collection \( F_i \) of incompressible surfaces with Betti numbers \( b_1F_i < b \) for all \( i \), none of which is a boundary parallel annulus or a boundary parallel disk, and no two of which are parallel, must have fewer than \( c \) members. Our estimate for \( c \) is exponential in \( b \). This theorem is used to detect closed incompressible surfaces in the infinite cyclic covers of all non-fibered knot complements. In other terms, if the commutator subgroup of a knot group is locally free, then it is actually a finitely generated free group.