Pin(2)-equivariant Seiberg–Witten Floer homology of Seifert fibrations

We compute the Pin(2) -equivariant Seiberg–Witten Floer homology of Seifert rational homology three-spheres in terms of their Heegaard Floer homology. As a result of this computation, we prove Manolescu’s conjecture that [STIX]x1D6FD=-[STIX]x1D707 for Seifert integral homology three-spheres. We show that the Manolescu invariants [STIX]x1D6FC,[STIX]x1D6FD, and [STIX]x1D6FE give new obstructions to homology cobordisms between Seifert fiber spaces, and that many Seifert homology spheres [STIX]x1D6F4(a₁,… ,aₙ) are not homology cobordant to any -[STIX]x1D6F4(b₁,… ,bₙ) . We then use the same invariants to give an example of an integral homology sphere not homology cobordant to any Seifert fiber space. We also show that the Pin(2) -equivariant Seiberg–Witten Floer spectrum provides homology cobordism obstructions distinct from [STIX]x1D6FC,[STIX]x1D6FD, and [STIX]x1D6FE . In particular, we identify an F[U] -module called connected Seiberg–Witten Floer homology, whose isomorphism class is a homology cobordism invariant.