An Algebraic Determination of Closed Orientable 3-Manifolds
William Jaco, Robert Cobb Myers · Transactions of the American Mathematical Society · 1979
Associated with each polyhedral simple closed curve j in a closed, orientable 3-manifold M is the fundamental group of the complement otj in M, v^M -j).The set, 3C(Ai), of knot groups of M is the set of groups vx(M -j) asj ranges over all polyhedral simple closed curves inM.We prove that two closed, orientable 3-manif olds M and N are homeomorphic if and only if 3C(M) = %(N).We refine the set of knot groups to a subset 9(M) of fibered knot groups of M and modify the above proof to show that two closed, orientable 3-manif olds M and N are homeomorpbic if and only if ^(M) -9(N).Associated with each polyhedral simple closed curve j in a closed orientable 3-manifold M is the fundamental group of the complement of j in M, irx(M -j).Hence, any closed orientable 3-manifold M has associated with it a set of groups %(M) defined to be precisely the groups irx(M -j) as j ranges over all polyhedral simple closed curves in M. The set of groups %(M) is the set of knot-groups of M.It was proposed by R. H. Fox at the Princeton Bicentennial Conference of 1946 [7, p. 24] that certain 3-manif olds may be distinguished by their knot-groups.In fact, Fox used this method [8] to reprove the PL-classification of lens spaces, established earlier by K. Reidemeister [17].E. J. Brody [3] extended the methods of Fox to establish a topological classification of lens spaces without reference to the Hauptvermutung, as well as a topological classification of the connected sum of two lens spaces.In this paper we prove that closed orientable 3-manif olds are topologically determined by their knot-groups.This is our Theorem 6.1 which states that two closed orientable 3-manifolds M and N axe homeomorphic if and only if %(M) = %(N).At the Georgia Topology Conference in 1969, A. C. Conner announced that if N is a homotopy 3-sphere and %(N) = %(S3), then N is homeomorphic to S3.The following year Conner announced, via an abstract in the Notices [5], the result that we prove in Theorem 6.1.He also circulated a