Monopole Floer homology and SOLV geometry

Lin, Francesco

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

On étudie l’homologie de Floer des monopôles d’une sphère d’homologie rationnelle $Y$ de type $\mathsf{Solv}$ du point de vue de la théorie spectrale. En appliquant des idées d’analyse de Fourier sur les groupes résolubles, on montre que pour des métriques $\mathsf{Solv}$ convenables sur $Y$, les petites perturbations régulières des équations de Seiberg–Witten n’admettent pas de solutions irréductibles ; en particulier ceci fournit une preuve géométrique du fait que $Y$ n’est pas un $L$-espace.

We study the monopole Floer homology of a $\mathsf{Solv}$ 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 $\mathsf{Solv}$ 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.

Classification : 57M27, 57R58, 58J50
Mots clés : Floer homology, Seiberg–Witten equations, Solvmanifolds

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