On Seifert fibered spaces embedding in 4-space, bounding definite manifolds and quasi-alternating Montesinos links

This dissertation is concerned with the question of which Seifert fibered spaces smoothly embed in the 4-sphere and the related question of which Seifert fibered spaces bound both a positive definite and a negative definite smooth 4-manifold. Using Donaldson’s diagonalization theorem we derive strong obstructions in both of these settings. We construct new embeddings of Seifert fibered spaces in S⁴ out of old ones, giving many new examples of Seifert fibered spaces which embed in S⁴ . Our results allow us to classify precisely when a Seifert fibered space over an orientable base surface smoothly embeds in S⁴ provided e > k/2, where e is the normalized central weight and k is the number of singular fibers. Based on these results and an analysis of the Neumann-Siebenmann invariant [mu with macron], we make some conjectures concerning Seifert fibered spaces which embed in S⁴. Finally, we classify the quasi-alternating Montesinos links, showing that a Montesinos link L is quasialternating if and only if its double branched cover is an L-space which bounds definite manifolds of both signs with torsion-free first homology