Small Seifert Fibered Zero-Surgery

This thesis provides new results regarding which small Seifert fibered spaces arise as 0-surgery on a knot in the 3-sphere. We generalize a 0-surgery obstruction of Ozsváth-Szabó to the setting of involutive Heegaard Floer homology, an extension of Heegaard Floer homology due to Hendricks-Manolescu. Using this obstruction, we find a new infinite family of small Seifert fibered spaces with first homology the integers and weight 1 fundamental group that cannot be obtained by 0-surgery on a knot in the 3-sphere, generalizing a result of Hedden-Kim-Mark-Park. In fact, we show that these manifolds cannot even be the boundary of a negative semi-definite spin 4-manifold. On the opposite end of the spectrum, we also provide a new family of small Seifert fibered spaces that do arise as 0-surgery on a knot in the 3-sphere. This is a simple generalization of work of Ichihara-Motegi-Song. Additionally, we establish some constraints on the types of knots that can have small Seifert fibered 0-surgery.