The trace embedding lemma and spinelessness

TL;DR

This paper applies the trace embedding lemma to explore exotic phenomena in 4-manifolds, revealing differences in smooth structures and properties of topological surfaces, with implications for 4-manifold topology.

Contribution

It introduces new examples of 4-manifolds with distinct smooth structures and properties, demonstrating the power of the trace embedding lemma in 4-dimensional topology.

Findings

Existence of infinitely many pairs of homeomorphic 4-manifolds with different smooth properties

Construction of spineless 4-manifolds with unique features

Topological surfaces in 4-manifolds cannot always be approximated by PL surfaces

Abstract

We demonstrate new applications of the trace embedding lemma to the study of piecewise-linear surfaces and the detection of exotic phenomena in dimension four. We provide infinitely many pairs of homeomorphic 4-manifolds $W$ and $W'$ homotopy equivalent to $S^2$ which have smooth structures distinguished by several formal properties: $W'$ is diffeomorphic to a knot trace but $W$ is not, $W'$ contains $S^2$ as a smooth spine but $W$ does not even contain $S^2$ as a piecewise-linear spine, $W'$ is geometrically simply connected but $W$ is not, and $W'$ does not admit a Stein structure but $W$ does. In particular, the simple spineless 4-manifolds $W$ provide an alternative to Levine and Lidman's recent solution to Problem 4.25 in Kirby's list. We also show that all smooth 4-manifolds contain topological locally flat surfaces that cannot be approximated by piecewise-linear surfaces.