Pseudo-isotopies and embedded surfaces in 4-manifolds

Abstract

The focus of this thesis is the study of smooth 4-dimensional manifolds. We examine two problems relating to 4-manifolds, the first pertaining to pseudo-isotopies and diffeomorphisms of 4-manifolds, and the second pertaining to embedded surfaces in 4-manifolds. We summarise our key results below.

A diffeomorphism of a compact manifold is pseudo-isotopic to the identity if there is a diffeomorphism of which restricts to on , and which restricts to the identity on and . We construct examples of diffeomorphisms of 4-manifolds which are pseudo-isotopic but not isotopic to the identity. To do so, we further understanding of which elements of the ``second pseudo-isotopy obstruction'', defined by Hatcher and Wagoner, can be realised by pseudo-isotopies of 4-manifolds. We also prove that all elements of the first and second pseudo-isotopy obstructions can be realised after connected sums with copies of .

If and are homotopic embedded surfaces in a -manifold then they may be related by a regular homotopy (at the expense of introducing double points) or by a sequence of stabilisations and destabilisations (at the expense of adding genus). This naturally gives rise to two integer-valued notions of distance between the embeddings: the singularity distance and the stabilisation distance . We use techniques similar to those used by Gabai in his proof of the 4-dimensional light-bulb theorem, to prove that .