Pin(2)-equivariant Seiberg–Witten Floer homology of Seifert fibrations. (9th February 2020) - Stoffregen, Matthew
Abstract : We compute the $\text{Pin}(2)$ -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 $\unicode[STIX]{x1D6FD}=-\bar{\unicode[STIX]{x1D707}}$ for Seifert integral homology three-spheres. We show that the Manolescu invariants $\unicode[STIX]{x1D6FC}, \unicode[STIX]{x1D6FD}, $ and $\unicode[STIX]{x1D6FE}$ give new obstructions to homology cobordisms between Seifert fiber spaces, and that many Seifert homology spheres $\unicode[STIX]{x1D6F4}(a_{1}, …, a_{n})$ are not homology cobordant to any $-\unicode[STIX]{x1D6F4}(b_{1}, …, b_{n})$ . 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 $\text{Pin}(2)$ -equivariant Seiberg–Witten Floer spectrum provides homology cobordism obstructions distinct from $\unicode[STIX]{x1D6FC}, \unicode[STIX]{x1D6FD}, $ and $\unicode[STIX]{x1D6FE}$ . In particular, we identify an $\mathbb{F}[U]$ -module called connected Seiberg–Witten Floer homology, whose isomorphism class is a homology cobordism invariant.