The Michigan Mathematical Journal

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

Irving Dai

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We compute the 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 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 Pin(2)-homology as an Abelian group. As an application, we show that β(Y,s)=μ¯(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 μ¯ 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

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

Received: 18 November 2016
Revised: 1 February 2017
First available in Project Euclid: 12 April 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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.

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