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.
[BGV04] Heat kernels and Dirac operators, Grundlehren Text Editions, Springer, 2004 | Zbl 1037.58015
[KLT11] HF=HM I : Heegaard Floer homology and Seiberg–Witten Floer homology (2011) (arXiv:math/https://arxiv.org/abs/1007.1979)
[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)