Absolutely exotic compact 4-manifolds (Q268273)

The authors show how to construct absolutely exotic smooth structures on compact 4-manifolds with boundary.NEWLINENEWLINEA smooth manifold \(W\) admits a relatively exotic smooth structure if there is a diffeomorphism \(f: \partial W\to\partial W\) which extends to a self-homeomorphism of \(W\) but not to a self-diffeomorphism. It is proved that any smooth compact 4-manifold \(W\) that admits a relatively exotic smooth structure contains a pair of codimension-zero submanifolds that are absolutely exotic copies of each other, i.e. they are homeomorphic but not diffeomorphic to each other. Moreover, they are homotopy equivalent to \(W\).NEWLINENEWLINEApplying this technique to corks discovered by \textit{S. Akbulut} [J. Differ. Geom. 33, No. 2, 335--356 (1991; Zbl 0839.57015)], i.e. compact smooth contractible manifolds which admit a relatively exotic structure, yields examples of absolutely exotic smooth structures on contractible \(4\)-manifolds. So far the smallest known absolutely exotic manifold with boundary was homotopy equivalent to \(S^2\) [\textit{S. Akbulut}, J. Differ. Geom. 33, No. 2, 357--361 (1991; Zbl 0839.57016)].NEWLINENEWLINEThe examples are constructed by adding an invertible homology cobordism along the boundary of a relatively exotic manifold.NEWLINENEWLINEThe authors also give explicit examples of absolutely exotic corks and anti-corks and show that the existence of infinitely many relatively exotic contractible \(4\)-manifolds implies the existence of infinitely many absolutely exotic ones.