Heegaard gradient and virtual fibers.
We show that any Heegaard splitting of the pair of the solid torus (≅D2xS1) and its core loop (an interior point xS1) is standard, using the notion of Heegaard splittings of pairs of 3-manifolds and properly imbedded graphs, which is defined in this paper.
A homology lens space is a closed 3-manifold with ℤ-homology groups isomorphic to those of a lens space. A useful theorem found in [Fu] states that a homology lens space may be obtained by an (n/1)-Dehn surgery on a homology 3-sphere if and only if the linking form of is equivalent to (1/n). In this note we generalize this result to cover all homology lens spaces, and in the process offer an alternative proof based on classical 3-manifold techniques.