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

TL;DR

This paper computes the Pin(2)-equivariant Seiberg-Witten Floer homology for Seifert rational homology spheres, proving a conjecture, introducing new cobordism obstructions, and identifying a new invariant related to homology cobordism.

Contribution

It provides explicit computations of Pin(2)-equivariant Floer homology for Seifert spheres, proves Manolescu's conjecture, and introduces new invariants for homology cobordism obstructions.

Findings

Proves Manolescu's conjecture for Seifert spheres.

Identifies new obstructions to homology cobordisms.

Defines a new homology cobordism invariant, connected Seiberg-Witten Floer homology.

Abstract

We compute the $\mathrm{Pin}(2)$-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 $\beta=-\bar{\mu}$ for Seifert integral homology three-spheres. We show that the Manolescu invariants $\alpha, \beta,$ and $\gamma$ give new obstructions to homology cobordisms between Seifert fiber spaces, and that many Seifert homology spheres $\Sigma(a_1,...,a_n)$ are not homology cobordant to any $-\Sigma(b_1,...,b_n)$. 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 $\mathrm{Pin}(2)$-equivariant Seiberg-Witten Floer spectrum provides homology cobordism obstructions distinct from $\alpha,\beta,$ and $\gamma$. In particular, we identify an $\mathbb{F}[U]$-module called connected Seiberg-Witten Floer homology, whose isomorphism class is a homology cobordism invariant.