Non-smoothable homeomorphisms of 4-manifolds

Galvin, Daniel A.P. (2024) Non-smoothable homeomorphisms of 4-manifolds. PhD thesis, University of Glasgow.

Abstract

We say that a homeomorphism f: X → X′ between two smooth manifolds is nonsmoothable if it is not isotopic to any diffeomorphism. We produce many different examples of non-smoothable homeomorphisms of various subtleties, and discuss their properties. We show that there is a one-to-one correspondence between such nonsmoothable homeomorphisms and diffeomorphic but not isotopic smooth structures on 4-manifolds, and we give an explicit construction of an infinite family of diffeomorphic but not isotopic smooth structures on the K3-surface.

In joint work with Roberto Ladu, we produce the first examples of non-smoothable homeomorphisms of simply-connected 4-manifolds such that the homeomorphism acts trivially on the homology of the manifold. The non-smoothability of these homeomorphisms is detected using gauge theory, and is unstable in the sense that these homeomorphisms become smoothable after sufficiently connected-summing with S² × S². We use this fact to create interesting diffeomorphisms of simply-connected 4-manifolds with boundary which act trivially on the homology of the manifold, but do not arise by inserting a loop of diffeomorphisms into the collar of the boundary. This ends the joint work.

A key focus of this work is on the Casson-Sullivan invariant cs(f) ∈ H³(X;Z/2) of a homeomorphism f: X → X′, which is the obstruction to a homeomorphism being stably pseudo-isotopic to a diffeomorphism. In particular, if a homeomorphism has non-trivial Casson-Sullivan invariant then it is non-smoothable, even after connectedsumming with S² × S². This invariant has a distinctly different flavour to that of gauge-theoretic invariants, and is more in-line with high-dimensional smoothing theory. Using surgery theory, we realise this invariant in a number of contexts. Firstly, we realise it for any orientable 4-manifold after a single connected-sum with S² × S². Secondly, we realise it unstably for many examples of 4-manifolds, including those with finite cyclic fundamental group.

We also discuss two applications of our work on the Casson-Sullivan invariant. The first application is to embedded surfaces, and we prove that any two topologically isotopic, smoothly embedded surfaces in a simply-connected 4-manifold become smoothly isotopic after externally connected-summing with S² × S² away from the surfaces.

The second application is to 3-manifolds. Let Y be a smooth 3-manifold. We consider the inclusion induced map Diff(Y ) → Homeo(Y) between the block diffeomorphism and block homeomorphism spaces, which are defined as geometric realisations of simplicial spaces and whose connected components correspond to the smooth and topological pseudo-mapping class groups, respectively. These spaces contain the classical spaces Diff(Y ) and Homeo(Y ) as subspaces. We show that for certain types of elliptic 3-manifolds, the inclusion induced maps Diff(Y ) → Homeo(Y ) and Homeo(Y) → Homeo(Y) are not 1-connected.

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Subjects: Q Science > QA Mathematics
• Colleges/Schools: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Supervisor's Name: Powell, Professor Mark and Owens, Professor Brendan
• Date of Award: 2024
• Depositing User: Theses Team
• Unique ID: glathesis:2024-84595
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 27 Sep 2024 14:27
• Last Modified: 27 Sep 2024 14:30
• Thesis DOI: 10.5525/gla.thesis.84595
• URI: https://theses.gla.ac.uk/id/eprint/84595