Higher order corks

Abstract

It is shown that any finite list of smooth, closed, simply-connected 4-manifolds that are homeomorphic to a given one X can be obtained by removing a single compact contractible submanifold (or cork) from X, and then regluing it by powers of a boundary diffeomorphism. Furthermore, by allowing the cork to be noncompact, the collection of all the smooth manifolds homeomorphic to X can be obtained in this way. The existence of a universal noncompact cork is also established. † If X C,h and X are diffeomorphic, the embedding C ⊂ X is said to be trivial. The cork (C, h) is trivial if all its embeddings in all 4-manifolds are trivial, or equivalently if h extends to a diffeomorphism of C [4].

Particulate exotica

“…Plugs tend to appear whenever corks 29 Or glue back something different but share the same boundary thus also slots into the open socket. 30 While higher order periodic diffeomorphisms are possible [37], involutions are sufficient for exoticity, e.g., [38] only utilizes involutions. 31 That is, it is only relatively exotic -it doesn't lack a diffeomorphism into the standard smoothness structure, it just lacks one that restricts exactly into the prescribed involution on the boundary.…”