Geometry & Topology

Floer homology and covering spaces

Tye Lidman and Ciprian Manolescu

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/gt.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We prove a Smith-type inequality for regular covering spaces in monopole Floer homology. Using the monopole Floer/Heegaard Floer correspondence, we deduce that if a 3–manifold Y admits a pn–sheeted regular cover that is a pL–space (for p prime), then Y is a pL–space. Further, we obtain constraints on surgeries on a knot being regular covers over other surgeries on the same knot, and over surgeries on other knots.

Article information

Source
Geom. Topol., Volume 22, Number 5 (2018), 2817-2838.

Dates
Received: 12 February 2017
Accepted: 5 November 2017
First available in Project Euclid: 26 March 2019

Permanent link to this document
https://projecteuclid.org/euclid.gt/1553565673

Digital Object Identifier
doi:10.2140/gt.2018.22.2817

Mathematical Reviews number (MathSciNet)
MR3811772

Zentralblatt MATH identifier
1395.57041

Subjects
Primary: 57R58: Floer homology
Secondary: 57M10: Covering spaces 57M60: Group actions in low dimensions

Keywords
Smith inequality Seiberg–Witten Heegaard Floer homology virtually cosmetic L–spaces

Citation

Lidman, Tye; Manolescu, Ciprian. Floer homology and covering spaces. Geom. Topol. 22 (2018), no. 5, 2817--2838. doi:10.2140/gt.2018.22.2817. https://projecteuclid.org/euclid.gt/1553565673


Export citation

References

  • I Agol, The virtual Haken conjecture, Doc. Math. 18 (2013) 1045–1087
  • K L Baker, A H Moore, Montesinos knots, Hopf plumbings, and L-space surgeries, J. Math. Soc. Japan 70 (2018) 95–110
  • J M Bloom, A link surgery spectral sequence in monopole Floer homology, Adv. Math. 226 (2011) 3216–3281
  • S Boyer, A Clay, Foliations, orders, representations, L-spaces and graph manifolds, Adv. Math. 310 (2017) 159–234
  • S Boyer, C M Gordon, L Watson, On L-spaces and left-orderable fundamental groups, Math. Ann. 356 (2013) 1213–1245
  • S Boyer, D Rolfsen, B Wiest, Orderable $3$–manifold groups, Ann. Inst. Fourier $($Grenoble$)$ 55 (2005) 243–288
  • G E Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics 46, Academic, New York (1972)
  • V Colin, P Ghiggini, K Honda, The equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions, I, preprint (2012)
  • V Colin, P Ghiggini, K Honda, The equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions, II, preprint (2012)
  • V Colin, P Ghiggini, K Honda, The equivalence of Heegaard Floer homology and embedded contact homology, III: From hat to plus, preprint (2012)
  • C Conley, Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Mathematics 38, Amer. Math. Soc., Providence, RI (1978)
  • A Floer, A refinement of the Conley index and an application to the stability of hyperbolic invariant sets, Ergodic Theory Dynam. Systems 7 (1987) 93–103
  • E E Floyd, On periodic maps and the Euler characteristics of associated spaces, Trans. Amer. Math. Soc. 72 (1952) 138–147
  • C M Gordon, Dehn surgery on knots, from “Proceedings of the International Congress of Mathematicians” (I Satake, editor), volume I, Math. Soc. Japan, Tokyo (1991) 631–642
  • J Hanselman, J Rasmussen, S D Rasmussen, L Watson, Taut foliations on graph manifolds, preprint (2015)
  • K Hendricks, A rank inequality for the knot Floer homology of double branched covers, Algebr. Geom. Topol. 12 (2012) 2127–2178
  • J Hom, A note on cabling and $L$–space surgeries, Algebr. Geom. Topol. 11 (2011) 219–223
  • S Jabuka, Heegaard Floer groups of Dehn surgeries, J. Lond. Math. Soc. 92 (2015) 499–519
  • S Jabuka, T E Mark, On the Heegaard Floer homology of a surface times a circle, Adv. Math. 218 (2008) 728–761
  • P Kronheimer, T Mrowka, Monopoles and three-manifolds, New Mathematical Monographs 10, Cambridge Univ. Press (2007)
  • P Kronheimer, T Mrowka, P Ozsváth, Z Szabó, Monopoles and lens space surgeries, Ann. of Math. 165 (2007) 457–546
  • C Kutluhan, Y-J Lee, C H Taubes, $\mathrm{HF}=\mathrm{HM}$, I: Heegaard Floer homology and Seiberg–Witten Floer homology, preprint (2010) To appear in Geom. Topol.
  • C Kutluhan, Y-J Lee, C H Taubes, $\mathrm{HF}=\mathrm{HM}$, II: Reeb orbits and holomorphic curves for the ech/Heegaard–Floer correspondence, preprint (2010) To appear in Geom. Topol.
  • C Kutluhan, Y-J Lee, C H Taubes, $\mathrm{HF}=\mathrm{HM}$, III: Holomorphic curves and the differential for the ech/Heegaard Floer correspondence, preprint (2010) To appear in Geom. Topol.
  • C Kutluhan, Y-J Lee, C H Taubes, $\mathrm{HF}=\mathrm{HM}$, IV: The Seiberg–Witten Floer homology and ech correspondence, preprint (2011) To appear in Geom. Topol.
  • C Kutluhan, Y-J Lee, C H Taubes, $\mathrm{HF}=\mathrm{HM}$, V: Seiberg–Witten–Floer homology and handle addition, preprint (2012) To appear in Geom. Topol.
  • D A Lee, R Lipshitz, Covering spaces and $\mathbb Q$–gradings on Heegaard Floer homology, J. Symplectic Geom. 6 (2008) 33–59
  • Y-J Lee, Heegaard splittings and Seiberg–Witten monopoles, from “Geometry and topology of manifolds” (H U Boden, I Hambleton, A J Nicas, B D Park, editors), Fields Inst. Commun. 47, Amer. Math. Soc., Providence, RI (2005) 173–202
  • T Lidman, C Manolescu, The equivalence of two Seiberg–Witten Floer homologies, preprint (2016)
  • T Lidman, A H Moore, Pretzel knots with $L$–space surgeries, Michigan Math. J. 65 (2016) 105–130
  • R Lipshitz, D Treumann, Noncommutative Hodge-to-de Rham spectral sequence and the Heegaard Floer homology of double covers, J. Eur. Math. Soc. 18 (2016) 281–325
  • P Lisca, A I Stipsicz, Ozsváth–Szabó invariants and tight contact $3$–manifolds, III, J. Symplectic Geom. 5 (2007) 357–384
  • C Manolescu, Seiberg–Witten–Floer stable homotopy type of three-manifolds with $b_1=0$, Geom. Topol. 7 (2003) 889–932
  • L Moser, Elementary surgery along a torus knot, Pacific J. Math. 38 (1971) 737–745
  • W D Neumann, D Zagier, Volumes of hyperbolic three-manifolds, Topology 24 (1985) 307–332
  • Y Ni, Z Wu, Cosmetic surgeries on knots in $S^3$, J. Reine Angew. Math. 706 (2015) 1–17
  • P Ozsváth, Z Szabó, Holomorphic disks and genus bounds, Geom. Topol. 8 (2004) 311–334
  • P Ozsváth, Z Szabó, Holomorphic disks and three-manifold invariants: properties and applications, Ann. of Math. 159 (2004) 1159–1245
  • P Ozsváth, Z Szabó, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. 159 (2004) 1027–1158
  • P Ozsváth, Z Szabó, On knot Floer homology and lens space surgeries, Topology 44 (2005) 1281–1300
  • P Ozsváth, Z Szabó, On the Heegaard Floer homology of branched double-covers, Adv. Math. 194 (2005) 1–33
  • P S Ozsváth, Z Szabó, Knot Floer homology and integer surgeries, Algebr. Geom. Topol. 8 (2008) 101–153
  • P S Ozsváth, Z Szabó, Knot Floer homology and rational surgeries, Algebr. Geom. Topol. 11 (2011) 1–68
  • T Peters, On $L$–spaces and non left-orderable $3$–manifold groups, preprint (2009)
  • A M Pruszko, The Conley index for flows preserving generalized symmetries, from “Conley index theory” (K Mischaikow, M Mrozek, P Zgliczyński, editors), Banach Center Publ. 47, Polish Acad. Sci. Inst. Math., Warsaw (1999) 193–217
  • P Seidel, I Smith, Localization for involutions in Floer cohomology, Geom. Funct. Anal. 20 (2010) 1464–1501
  • P A Smith, Transformations of finite period, Ann. of Math. 39 (1938) 127–164
  • C H Taubes, Embedded contact homology and Seiberg–Witten Floer cohomology, I, Geom. Topol. 14 (2010) 2497–2581
  • C H Taubes, Embedded contact homology and Seiberg–Witten Floer cohomology, II, Geom. Topol. 14 (2010) 2583–2720
  • C H Taubes, Embedded contact homology and Seiberg–Witten Floer cohomology, III, Geom. Topol. 14 (2010) 2721–2817
  • C H Taubes, Embedded contact homology and Seiberg–Witten Floer cohomology, IV, Geom. Topol. 14 (2010) 2819–2960
  • C H Taubes, Embedded contact homology and Seiberg–Witten Floer cohomology, V, Geom. Topol. 14 (2010) 2961–3000
  • D T Wise, The structure of groups with a quasiconvex hierarchy, preprint (2011) Available at \setbox0\makeatletter\@url http://www.math.mcgill.ca/wise/papers.html {\unhbox0