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 has constructed uncountably many smoothings of R^4 which do support Stein structures.) Other applications of this invariant are given.
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