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May 2018 On the Pin(2)-Equivariant Monopole Floer Homology of Plumbed 3-Manifolds
Irving Dai
Michigan Math. J. 67(2): 423-447 (May 2018). DOI: 10.1307/mmj/1523498585

Abstract

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.

Citation

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Irving Dai. "On the Pin(2)-Equivariant Monopole Floer Homology of Plumbed 3-Manifolds." Michigan Math. J. 67 (2) 423 - 447, May 2018. https://doi.org/10.1307/mmj/1523498585

Information

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

zbMATH: 06914769
MathSciNet: MR3802260
Digital Object Identifier: 10.1307/mmj/1523498585

Subjects:
Primary: 57M27, 57R58

Rights: Copyright © 2018 The University of Michigan

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Vol.67 • No. 2 • May 2018
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