Exotic smoothings via large R⁴’s in Stein surfaces
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We study the relationship between exotic R⁴'s and Stein surfaces as it applies to smoothing theory on more general open 4–manifolds. In particular, we construct the first known examples of large exotic R⁴'s that embed in Stein surfaces. This relies on an extension of Casson's Embedding Theorem for locating Casson handles in closed 4–manifolds. Under sufficiently nice conditions, we show that using these R⁴'s as end-summands produces uncountably many diffeomorphism types while maintaining independent control over the genus-rank function and the Taylor invariant. Additionally, we prove that a family of knots known to be topologically slice but not smoothly slice is rationally slice, realizing a combination of slice properties that seems to have been previously unknown to exist.