## The Michigan Mathematical Journal

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

Irving Dai

#### Abstract

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.

#### Article information

Source
Michigan Math. J., Volume 67, Issue 2 (2018), 423-447.

Dates
Revised: 1 February 2017
First available in Project Euclid: 12 April 2018

https://projecteuclid.org/euclid.mmj/1523498585

Digital Object Identifier
doi:10.1307/mmj/1523498585

Mathematical Reviews number (MathSciNet)
MR3802260

Zentralblatt MATH identifier
06914769

Subjects
Primary: 57R58: Floer homology 57M27: Invariants of knots and 3-manifolds

#### Citation

Dai, Irving. On the $\operatorname{Pin}(2)$ -Equivariant Monopole Floer Homology of Plumbed 3-Manifolds. Michigan Math. J. 67 (2018), no. 2, 423--447. doi:10.1307/mmj/1523498585. https://projecteuclid.org/euclid.mmj/1523498585

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