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This paper studies the class of spherical objects over any Kodaira $n$-cycleof projective lines and provides a parametrization of their isomorphism classesin terms of closed curves on the $n$-punctured torus withoutself-intersections. Employing recent results on gentle algebras, we derive atopological model for the bounded derived category of any Kodaira cycle. Thegroups of triangle auto-equivalences of these categories are computed and areshown to act transitively on isomorphism classes of spherical objects. Thisanswers a question by Polishchuk and extends earlier results byBurban-Kreussler and Lekili-Polishchuk. The description of auto-equivalences isfurther used to establish faithfulness of a mapping class group action definedby Sibilla. The final part describes the closed curves which correspond tovector bundles and simple vector bundles. This leads to an alternative proof ofa result by Bodnarchuk-Drozd-Greuel which states that simple vector bundles oncycles of projective lines are uniquely determined by their multi-degree, rankand determinant. As a by-product we obtain a closed formula for the cyclicsequence of any simple vector bundle on $C_n$ as introduced byBurban-Drozd-Greuel.

Spherical objects, transitivity and auto-equivalences of Kodaira cycles via gentle algebras