On the $\operatorname{Pin}(2)$ -Equivariant Monopole Floer Homology of Plumbed 3-Manifolds

We compute the $\operatorname{Pin}(2)$ -equivariant monopole Floer homology for the class of plumbed 3-manifolds considered by Ozsváth and Szabó [18]. We show that for these manifolds, the $\operatorname{Pin}(2)$ -equivariant monopole Floer homology can be calculated in terms of the Heegaard Floer/monopole Floer lattice complex defined by Némethi [15]. 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 $\operatorname{Pin}(2)$ -homology as an Abelian group. As an application, we show that $\beta(-Y,s)=\bar{\mu}(Y,s)$ for all plumbed 3-manifolds with at most one “bad” vertex, proving (an analogue of) a conjecture posed by Manolescu [12]. Our proof also generalizes results by Stipsicz [21] and Ue [26] relating $\bar{\mu}$ with the Ozsváth–Szabó $d$ -invariant. Some observations aimed at extending our computations to manifolds with more than one bad vertex are included at the end of the paper.