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, ${\pi _1}(M - j)$. The set, $\mathcal {K}(M)$, of knot groups of M is the set of groups ${\pi _1}(M - j)$ as j ranges over all polyhedral simple closed curves in M. We prove that two closed, orientable 3-manifolds M and N are homeomorphic if and only if $\mathcal {K}(M) = \mathcal {K}(N)$. We refine the set of knot groups to a subset $\mathcal {F}(M)$ 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 $\mathcal {F}(M) = \mathcal {F}(N)$.