Abstract

We study Manolescu's $\mathrm{Pin}$-equivariant Seiberg-Witten Floer homology of rational homology three-spheres, with applications to the homology cobordism group $\theta^H_3$ in mind. We compute this homology theory for Seifert rational homology three-spheres in terms of their Heegaard Floer homology. We prove Manolescu's conjecture that $\beta=-\bar{\mu}$, the Neumann-Siebenmann invariant, for Seifert integral homology three-spheres. We establish the existence of integral homology spheres not homology cobordant to any Seifert space. We show that there is a naturally defined subgroup of the homology cobordism group, generated by certain Seifert spaces, which admits a $\mathbb{Z}^\infty$ summand, generalizing the theorem of Fintushel-Stern and Furuta on the infinite-generation of the homology cobordism group. In addition to the application of the $\mathrm{Pin}$-theory to Seifert spaces, we apply it to the full homology cobordism group. In this direction, we identify a $\mathbb{F}[U]$-submodule of Heegaard Floer homology, called connected Seiberg-Witten Floer homology, whose isomorphism class is a homology cobordism invariant.