On the topology of Lagrangian fillings of the standard Legendrian sphere

In this paper we study the uniqueness of Lagrangian fillings of the standard Legendrian sphere L₀ in the standard contact sphere (S²ⁿ⁻¹, ξ_st ). We show that every exact Maslov zero Lagrangian filling L of L₀ in a Liouville filling of (S²ⁿ⁻¹, ξ_st ) is a homology ball. If we restrict ourselves to real Lagrangian fillings, then L is diffeomorphic to the n-ball for n ≥ 6.