The symplectic topology of some rational homology balls

TL;DR

This paper investigates the symplectic topology of certain rational homology balls derived from algebraic quotients of the An Milnor fibre, revealing the absence of closed exact Lagrangians and the presence of essential monotone Lagrangian tori, leading to non-vanishing symplectic cohomology.

Contribution

It demonstrates the non-existence of closed exact Lagrangian submanifolds and identifies essential monotone Lagrangian tori in these affine surfaces, advancing understanding through homological mirror symmetry.

Findings

No closed exact Lagrangian submanifolds exist in these surfaces.

Existence of Floer theoretically essential monotone Lagrangian tori.

Non-vanishing symplectic cohomology of the surfaces.

Abstract

We study the symplectic topology of some finite algebraic quotients of the An 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 An 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 An Milnor fibre. We conclude that these affine surfaces have non-vanishing symplectic cohomology.