On the Pin(2)-equivariant monopole Floer homology of plumbed 3-manifolds

TL;DR

This paper computes the Pin(2)-equivariant monopole Floer homology for certain plumbed 3-manifolds, linking it to Heegaard Floer invariants and confirming a conjecture relating correction terms and invariants.

Contribution

It provides a method to compute Pin(2)-equivariant monopole Floer homology for manifolds with at most one bad vertex, connecting it to existing lattice complexes and invariants.

Findings

Pin(2)-homology determined by monopole Floer ranks

Confirmed Manolescu's conjecture for these manifolds

Extended relations between invariants and $d$-invariants

Abstract

We compute the Pin(2)-equivariant monopole Floer homology for the class of plumbed 3-manifolds with at most one "bad" vertex (in the sense of Ozsvath and Szabo). We show that for these manifolds, the Pin(2)-equivariant monopole Floer homology can be calculated in terms of the Heegaard Floer/monopole Floer lattice complex defined by Nemethi. Moreover, we prove that in such cases the ranks of the usual monopole Floer homology groups suffice to determine both the Manolescu correction terms and the Pin(2)-homology as an abelian group. As an application of this, we show that $\beta(-Y, s) = \bar{\mu}(Y, s)$ for all plumbed 3-manifolds with at most one bad vertex, proving a conjecture posed by Manolescu. Our proof also generalizes results by Stipsicz and Ue relating the Neumann-Siebenmann invariant with the Ozsvath-Szabo $d$-invariant.