An algebraic determination of closed orientable 3-manifolds
Abstract
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, π₁(M — j). The set, K(M), of knot groups of M is the set of groups π₁(M — j) as j ranges over all polyhedral simple closed curves inM. We prove that two closed, orientable 3-manifolds M and N are homeomorphic if and only if K(M) = K(N). We refine the set of knot groups to a subset F(M) of fibered knot groups of M and modify the above proof to show that two closed, orientable 3-manifolds M and N are homeomorphic if and only if F(M) = F(N).
Citation
Jaco, W., Myers, R. (1979). An algebraic determination of closed orientable 3-manifolds. Transactions of the American Mathematical Society, 253(0), pp. 149-170. https://doi.org/10.1090/S0002-9947-1979-0536940-6