Diffeomorphisms of 4-manifolds with boundary and exotic embeddings

Abstract

We define family versions of the invariant of 4-manifolds with contact boundary due to Kronheimer and Mrowka and using this to detect exotic diffeomorhisms on 4-manifold with boundary. Further we developed some techniques that has been used to show existence of exotic codimension-2 submanifolds and exotic codimension-1 submanifolds in 4-manifolds.

The groups of diffeomorphisms and homeomorphisms of 4-manifolds with boundary

“…To put this in context, recall that there are qualitatively different types of exotic diffeomorphisms that a 4-manifold can support. The first type is Ruberman's first example of exotic diffeomorphism [45], and its variants detected by families Seiberg-Witten theory [10,23]. Another variant stems from Dehn twists along Seifert fibered 3-manifolds [26,31,25].…”

“…4 is the smallest dimension where exotic diffeomorphisms exist. After a long hiatus following Ruberman's pioneering work [45,46] exotic diffeomorphisms of 4-manifolds have attracted significant interest in recent studies through the advancement of families Seiberg-Witten theory [10,26,31,33,23,25]. One of the main results in Ruberman [46] was to prove that the group of (components of) exotic diffeomorphisms can be infinitely generated for some 4-manifolds.…”