Abstract
We show that for any positive integer [Formula: see text], there exist pairs of compact, contractible, Stein 4-manifolds and order [Formula: see text] self-diffeomorphisms of the boundaries that do not extend to the full manifolds. Each boundary of the Stein 4-manifolds is a cyclic branched cover along a slice knot embedded in the boundary of a contractible 4-manifold. Each pair is called a finite order cork, we give a method producing examples of many finite order corks, which are possibly not a Stein manifold. The example of the Stein cork gives a diffeomorphism generating [Formula: see text] homotopic but non-isotopic Stein fillable contact structures for an arbitrary positive integer [Formula: see text].
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