Metadata
Abstract
We study the monopole Floer homology of a rational homology sphere from the point of view of spectral theory. Applying ideas of Fourier analysis on solvable groups, we show that for suitable metrics on , small regular perturbations of the Seiberg–Witten equations do not admit irreducible solutions; in particular, this provides a geometric proof that is an -space.
References
[ADS83] Eta invariants, signature defects of cusps, and values of -functions, Ann. Math., Volume 118 (1983) no. 1, pp. 131-177 | Article | MR 707164 | Zbl 0531.58048
[APS75] Spectral asymmetry and Riemannian geometry. I, Math. Proc. Camb. Philos. Soc., Volume 77 (1975), pp. 43-69 | Article | MR 397797 | Zbl 0297.58008
[Bal08] 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] Heat kernels and Dirac operators, Grundlehren Text Editions, Springer, 2004 | Zbl 1037.58015
[BGW13] 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] Harmonic analysis on compact solvmanifolds, Lecture Notes in Mathematics, Volume 602, Springer, 1977, vii+179 pages | MR 447471 | Zbl 0403.43007
[CGH12] 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] Hilbert modular surfaces, Enseign. Math., Volume 19 (1973), pp. 183-281 | MR 393045 | Zbl 0285.14007
[KLT11] HF=HM I : Heegaard Floer homology and Seiberg–Witten Floer homology (2011) (arXiv:math/https://arxiv.org/abs/1007.1979)
[KM07] Monopoles and three-manifolds, New Mathematical Monographs, Volume 10, Cambridge University Press, 2007 | Article | MR 2388043 | Zbl 1158.57002
[KMOS07] Monopoles and lens space surgeries, Ann. Math., Volume 165 (2007) no. 2, pp. 457-546 | Article | MR 2299739 | Zbl 1204.57038
[Lin16] Lectures on monopole Floer homology, Proceedings of the Gökova Geometry-Topology Conference 2015 (2016), pp. 39-80 | MR 3526838 | Zbl 1357.57001
[Lin17] 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] The Seiberg–Witten equations and the length spectrum of hyperbolic three-manifolds (2018) (arXiv:math/1810.06346)
[Mar16] An introduction to Geometric Topology (2016) (arXiv:math/1610.02592)
[MOY97] 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] Floer simple manifolds and L-space intervals, Adv. Math., Volume 322 (2017), pp. 738-805 | Article | MR 3720808 | Zbl 1379.57024
[Sak85] Involutions on torus bundles over , Osaka J. Math., Volume 22 (1985) no. 1, pp. 163-185 | MR 785529 | Zbl 0571.57028
[Sco83] The geometries of -manifolds, Bull. Lond. Math. Soc., Volume 15 (1983) no. 5, pp. 401-487 | Article | MR 705527 | Zbl 0561.57001
[Tur97] Torsion invariants of -structures on -manifolds, Math. Res. Lett., Volume 4 (1997) no. 5, pp. 679-695 | Article | MR 1484699 | Zbl 0891.57019