On the Pin(2)-Equivariant Monopole Floer Homology of Plumbed 3-Manifolds

Abstract

We compute the $Pin\left(2\right)$-equivariant monopole Floer homology for the class of plumbed 3-manifolds considered by Ozsváth and Szabó [18]. We show that for these manifolds, the $Pin\left(2\right)$-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 $Pin\left(2\right)$-homology as an Abelian group. As an application, we show that $\beta (-Y,s)=\overline{\mu }(Y,s)$ 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 $\overline{\mu }$ 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.