On a volume invariant of 3-manifolds

Abstract

This paper investigates a real-valued topological invariant of 3-manifolds called topological volume. For a given 3-manifold M, the topological volume volt(M) is defined as the smallest volume of the complement of a (possibly empty) hyperbolic link in M. Various refinements of this invariant are given, asymptotically tight upper and lower bounds are determined, and all non-hyperbolic closed 3-manifolds with topological volume of at most 3.07 are classified. Moreover, it is shown that for all but finitely many cases, the volume minimiser in a lens space is obtained by Dehn filling one of the cusps of the complement of the Whitehead link or its sister manifold.