The Michigan Mathematical Journal

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

Irving Dai

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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.

Article information

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

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

Permanent link to this document
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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References

  • [1] V. Colin, P. Ghiggini, and K. Honda, The equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions I, 2012, http://arxiv.org/abs/1208.1074.
  • [2] K. Frøyshov, Monopole Floer homology for rational homology 3-spheres, Duke Math. J. 155 (2010), no. 3, 519–576.
  • [3] P. Kronheimer and T. Mrowka, Monopoles and three-manifolds, Cambridge University Press, Cambridge, 2007.
  • [4] P. Kronheimer, T. Mrowka, P. Ozsváth, and Z. Szabó, Monopoles and lens space surgeries, Ann. of Math. 165 (2007), no. 2, 457–546.
  • [5] C. Kutluhan, Y. Lee, and C. Taubes, HF$=$HM I: Heegaard Floer homology and Seiberg–Witten Floer homology, 2010, http://arxiv.org/abs/1007.1979.
  • [6] T. Lidman and C. Manolescu, The equivalence of two Seiberg–Witten Floer homologies, 2016, https://arxiv.org/abs/1603.00582.
  • [7] F. Lin, A Morse–Bott approach to monopole Floer homology and the triangulation conjecture, 2014, http://arxiv.org/abs/1404.4561.
  • [8] F. Lin, The surgery exact triangle in $\operatorname{Pin}(2)$-monopole Floer homology, 2015, https://arxiv.org/abs/1504.01993.
  • [9] F. Lin, $\operatorname{Pin}(2)$-monopole Floer homology, higher compositions and connected sums, 2016, https://arxiv.org/abs/1605.03137.
  • [10] C. Manolescu, Seiberg–Witten Floer stable homotopy type of three-manifolds with $b_{1}=0$, Geom. Topol. 7 (2003), 889–932.
  • [11] C. Manolescu, The Conley index, gauge theory, and triangulations, J. Fixed Point Theory Appl. 13 (2013), no. 2, 431–457.
  • [12] C. Manolescu, $\operatorname{Pin}(2)$-Equivariant Seiberg–Witten Floer homology and the triangulation conjecture, J. Amer. Math. Soc. 29 (2016), 147–176.
  • [13] A. Némethi, On the Ozsváth–Szabó invariant of negative definite plumbed 3-manifolds, Geom. Topol. 9 (2005), 991–1042.
  • [14] A. Némethi, Graded roots and singularities, Singularities in geometry and topology, pp. 394–463, World Sci. Publ., New Jersey, 2007.
  • [15] A. Némethi, Lattice cohomology of normal surface singularities, Publ. Res. Inst. Math. Sci. 44 (2008), 507–543.
  • [16] W. Neumann, An invariant of plumbed homology spheres, Lecture Notes in Math. 788 (1980), 125–144.
  • [17] P. Ozsváth, A. Stipsicz, and Z. Szabó, A spectral sequence on lattice homology, 2012, https://arxiv.org/abs/1206.1654.
  • [18] P. Ozsváth and Z. Szabó, On the Floer homology of plumbed three-manifolds, Geom. Topol. 7 (2003), 185–224.
  • [19] D. Ruberman and N. Saveliev, The $\bar{\mu}$-invariant of Seifert fibered homology spheres and the Dirac operator, Geom. Dedicata 154 (2011), 93–101.
  • [20] L. Siebenmann, On vanishing of the Rohlin invariant and nonfinitely amphichiral homology 3-spheres, Lecture Notes in Math. 788 (1980), 172–222.
  • [21] A. Stipsicz, On the $\bar{\mu}$-invariant of rational surface singularities, Proc. Amer. Math. Soc. 136 (2008), 3815–3823.
  • [22] M. Stoffregen, $\operatorname{Pin}(2)$-equivariant Seiberg–Witten Floer homology of Seifert fiberations, 2015, http://arxiv.org/abs/1505.03234.
  • [23] M. Stoffregen, Manolescu invariants of connected sums, 2015, https://arxiv.org/abs/1510.01286.
  • [24] C. Taubes, The Seiberg–Witten equations and the Weinstein conjecture, Geom. Topol. 11 (2007), 2117–2202.
  • [25] E. Tweedy, Heegaard Floer homology and several families of Brieskorn spheres, Topology Appl. 160 (2013), no. 4, 620–632.
  • [26] M. Ue, The Fukumoto–Furuta and the Ozsváth–Szabó invariants for spherical 3-manifolds, Algebraic topology—old and new, pp. 121–139, Polish Acad. Sci. Inst. Math, Warsaw, 2009.