Annales Henri Lebesgue

Lin, Francesco

Monopole Floer homology and SOLV geometry

Annales Henri Lebesgue, Volume 3 (2020) , pp. 1117-1131.

Metadata

KeywordsFloer 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 | Article | MR 707164 | Zbl 0531.58048

[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 | Article | MR 397797 | Zbl 0297.58008

[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 | Article | MR 2461862 | Zbl 1160.57009

[BGV04] Berline, Nicole; Getzler, Ezra; Vergne, Michèle Heat kernels and Dirac operators, Grundlehren Text Editions, Springer, 2004 | Zbl 1037.58015

[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 | Article | MR 3072799 | Zbl 1279.57008

[Bre77] Brezin, Jonathan Harmonic analysis on compact solvmanifolds, Lecture Notes in Mathematics, Volume 602, Springer, 1977, vii+179 pages | MR 447471 | Zbl 0403.43007

[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 1256.57020

[Hir73] Hirzebruch, Friedrich E. P. Hilbert modular surfaces, Enseign. Math., Volume 19 (1973), pp. 183-281 | MR 393045 | Zbl 0285.14007

[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, Volume 10, Cambridge University Press, 2007 | Article | MR 2388043 | Zbl 1158.57002

[KMOS07] Kronheimer, P.; Mrowka, Tomasz; Ozsváth, Peter; Szabó, Zoltán Monopoles and lens space surgeries, Ann. Math., Volume 165 (2007) no. 2, pp. 457-546 | Article | MR 2299739 | Zbl 1204.57038

[Lin16] Lin, Francesco Lectures on monopole Floer homology, Proceedings of the Gökova Geometry-Topology Conference 2015 (2016), pp. 39-80 | MR 3526838 | Zbl 1357.57001

[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 | Article | MR 1611061 | Zbl 0933.57030

[RR17] Rasmussen, Jacob; Rasmussen, Sarah Dean Floer simple manifolds and L-space intervals, Adv. Math., Volume 322 (2017), pp. 738-805 | Article | MR 3720808 | Zbl 1379.57024

[Sak85] Sakuma, Makoto Involutions on torus bundles over $S^{1}$ , Osaka J. Math., Volume 22 (1985) no. 1, pp. 163-185 | MR 785529 | Zbl 0571.57028

[Sco83] Scott, Peter The geometries of $3$ -manifolds, Bull. Lond. Math. Soc., Volume 15 (1983) no. 5, pp. 401-487 | Article | MR 705527 | Zbl 0561.57001

[Tur97] Turaev, Vladimir G. Torsion invariants of ${Spin}^{c}$ -structures on $3$ -manifolds, Math. Res. Lett., Volume 4 (1997) no. 5, pp. 679-695 | Article | MR 1484699 | Zbl 0891.57019