Journal Article

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

Stoffregen, Matthew

2020

Overview

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 $\\unicode[STIX]{x1D6F4}(a_{1},\\ldots ,a_{n})$ are not homology cobordant to any $-\\unicode[STIX]{x1D6F4}(b_{1},\\ldots ,b_{n})$ . 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.

• Publisher: Cambridge University Press
• Subject: Homology / Integrals / Invariants / Isomorphism / Obstructions / Spheres