  We show that simply connected contact manifolds that are subcritically Stein
fillable have a unique symplectically aspherical filling up to diffeomorphism.
Various extensions to manifolds with non-trivial fundamental group are
discussed. The proof rests on homological restrictions on symplectic fillings
derived from a degree-theoretic analysis of the evaluation map on a suitable
moduli space of holomorphic spheres. Applications of this homological result
include a proof that compositions of right-handed Dehn twists on Liouville
domains are of infinite order in the symplectomorphism group. We also derive
uniqueness results for subcritical Stein fillings up to homotopy equivalence
and, under some topological assumptions on the contact manifold, up to
diffeomorphism or symplectomorphism.
