Szabo, ZoltanDai, IrvingMathematics Department2019-11-052026-09-302019-11-052026-09-302019http://arks.princeton.edu/ark:/88435/dsp0102870z743https://theses-dissertations.princeton.edu/handle/88435/dsp0102870z743In this thesis, we present several results regarding the application of Heegaard Floer theory to the homology cobordism group. The majority of our work is concerned with giving a structural understanding of the involutive Heegaard Floer homology for linear combinations of Seifert fibered spaces. As an application, we show that if $Y$ is a linear combination of Seifert fibered homology spheres with $\mu(Y) = 1$, then $Y$ is not torsion in the homology cobordism group. We also discuss what can be said about the Pin(2)-equivariant monopole Floer homology of Seifert fibered spaces using our techniques. These results give a possible approach towards showing that Seifert fibered spaces do not generate the homology cobordism group.enFloer theoryLow-dimensional topologyTheoretical mathematicsAcademic dissertations (Ph.D.)