Volume 3 (2020)
Lin, Francesco
Monopole Floer homology and SOLV geometry
Annales Henri Lebesgue, Volume 3 (2020), pp. 1117-1131.

Metadata

DOI10.5802/ahl.56
Keywords Floer homology, Seiberg–Witten equations, Solvmanifolds

Abstract

We study the monopole Floer homology of a Solv rational homology sphere Y from the point of view of spectral theory. Applying ideas of Fourier analysis on solvable groups, we show that for suitable Solv metrics on Y, small regular perturbations of the Seiberg–Witten equations do not admit irreducible solutions; in particular, this provides a geometric proof that Y is an L-space.


References

[ADS83] Atiyah, Michael F.; Donnelly, Harold; Singer, Isadore M. Eta invariants, signature defects of cusps, and values of L-functions, Ann. Math., Volume 118 (1983) no. 1, pp. 131-177 | DOI | MR | Zbl

[APS75] Atiyah, Michael F.; Patodi, Vijay K.; Singer, Isadore M. Spectral asymmetry and Riemannian geometry. I, Math. Proc. Camb. Philos. Soc., Volume 77 (1975), pp. 43-69 | DOI | MR | Zbl

[Bal08] Baldwin, John A. Heegaard Floer homology and genus one, one-boundary component open books, J. Topol., Volume 1 (2008) no. 4, pp. 963-992 | DOI | MR | Zbl

[BGV04] Berline, Nicole; Getzler, Ezra; Vergne, Michèle Heat kernels and Dirac operators, Grundlehren Text Editions, Springer, 2004 | Zbl

[BGW13] Boyer, Steven; Gordon, Cameron McA.; Watson, Liam On L-spaces and left-orderable fundamental groups, Math. Ann., Volume 356 (2013) no. 4, pp. 1213-1245 | DOI | MR | Zbl

[Bre77] Brezin, Jonathan Harmonic analysis on compact solvmanifolds, Lecture Notes in Mathematics, 602, Springer, 1977, vii+179 pages | MR | Zbl

[CGH12] Colin, Vincent; Ghiggini, Paolo; Honda, Ko The equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions I (2012) (https://arxiv.org/abs/1208.1074) | Zbl

[Hir73] Hirzebruch, Friedrich E. P. Hilbert modular surfaces, Enseign. Math., Volume 19 (1973), pp. 183-281 | MR | Zbl

[KLT11] Kutluhan, Cagatay; Lee, Yi-Jen; Taubes, Clifford HF=HM I : Heegaard Floer homology and Seiberg–Witten Floer homology (2011) (arXiv:math/https://arxiv.org/abs/1007.1979)

[KM07] Kronheimer, Peter; Mrowka, Tomasz S. Monopoles and three-manifolds, New Mathematical Monographs, 10, Cambridge University Press, 2007 | DOI | MR | Zbl

[KMOS07] Kronheimer, P.; Mrowka, Tomasz; Ozsváth, Peter; Szabó, Zoltán Monopoles and lens space surgeries, Ann. Math., Volume 165 (2007) no. 2, pp. 457-546 | DOI | MR | Zbl

[Lin16] Lin, Francesco Lectures on monopole Floer homology, Proceedings of the Gökova Geometry-Topology Conference 2015, Gökova Geometry/Topology Conference (GGT), Gökova (2016), pp. 39-80 | MR | Zbl

[Lin17] Lin, Francesco Monopole Floer homology and the spectral geometry of three-manifolds (2017) (https://arxiv.org/abs/1705.08817, to appear in Communications in Analysis and Geometry)

[LL18] Lin, Francesco; Lipnowski, Michael The Seiberg–Witten equations and the length spectrum of hyperbolic three-manifolds (2018) (arXiv:math/1810.06346)

[Mar16] Martelli, Bruno An introduction to Geometric Topology (2016) (arXiv:math/1610.02592)

[MOY97] Mrowka, Tomasz S.; Ozsváth, Peter; Yu, Baozhen Seiberg–Witten monopoles on Seifert fibered spaces, Commun. Anal. Geom., Volume 5 (1997) no. 4, pp. 685-791 | DOI | MR | Zbl

[RR17] Rasmussen, Jacob; Rasmussen, Sarah Dean Floer simple manifolds and L-space intervals, Adv. Math., Volume 322 (2017), pp. 738-805 | DOI | MR | Zbl

[Sak85] Sakuma, Makoto Involutions on torus bundles over S 1 , Osaka J. Math., Volume 22 (1985) no. 1, pp. 163-185 | MR | Zbl

[Sco83] Scott, Peter The geometries of 3-manifolds, Bull. Lond. Math. Soc., Volume 15 (1983) no. 5, pp. 401-487 | DOI | MR | Zbl

[Tur97] Turaev, Vladimir G. Torsion invariants of Spin c -structures on 3-manifolds, Math. Res. Lett., Volume 4 (1997) no. 5, pp. 679-695 | DOI | MR | Zbl