An invariant of smooth 4–manifolds

Abstract

We define a diffeomorphism invariant of smooth 4–manifolds which we can estimate for many smoothings of ${ℝ}^4$ and other smooth 4–manifolds. Using this invariant we can show that uncountably many smoothings of ${ℝ}^4$ support no Stein structure. Gompf constructed uncountably many smoothings of ${ℝ}^4$ which do support Stein structures. Other applications of this invariant are given.