Spherical Seifert fibered spaces, knot surgeries, and Heegaard Floer homology.

摘要

Thanks to Wallace and Lickorish, we know that any 3-manifold can be obtained by surgery on a link. In 1971, Moser asked which of these manifolds can be obtained from surgery on a knot. On the other hand, Berge and then Dean et al. tried to determine which knots give rise to given types of 3-manifold, in particular lens spaces and Seifert fibered spaces. We use Heegaard Floer theory to investigate these two questions using a set of invariants for a 3-manifold and its associated torsion Spinc structures called the correction terms. These terms can be calculated combinatorially either from a plumbing description of the manifold or from a knot surgery description. We show that the correction terms provide an obstruction to spherical Seifert fibered spaces (other than lens spaces) being realized as knot surgeries. For those spaces with small first homology, we show the invariant is a complete obstruction; we give reasons why it should also be useful for those with larger homology.