The symplectic topology of some rational homology balls

Abstract

We study the symplectic topology of some finite algebraic quotients of the $A_n$ Milnor fibre which are diffeomorphic to the rational homology balls that appear in Fintushel and Stern’s rational blowdown construction. We prove that these affine surfaces have no closed exact Lagrangian submanifolds by using the already available and deep understanding of the Fukaya category of the $A_n$ Milnor fibre coming from homological mirror symmetry. On the other hand, we find Floer theoretically essential monotone Lagrangian tori, finitely covered by the monotone tori which we study in the $A_n$ Milnor fibre. We conclude that these affine surfaces have non-vanishing symplectic cohomology.