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We define a diffeomorphism invariant of smooth 4–manifolds which we
can estimate for many smoothings of R4 and other smooth
4–manifolds. Using this invariant we can show that uncountably
many smoothings of R4 support no Stein structure. Gompf
constructed uncountably many smoothings of R4 which do
support Stein structures. Other applications of this invariant are given.
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