Cobordism and Concordance of Surfaces in 4-Manifolds

TL;DR

This paper characterizes when two surfaces in 4-manifolds are cobordant or concordant, providing new constructions and classifications, especially for non-orientable surfaces in simply-connected 4-manifolds.

Contribution

It offers a complete classification of closed surfaces in simply-connected 4-manifolds up to concordance, extending Sunukjian's method to unoriented cases using Pin$^-$-structures.

Findings

Two surfaces are cobordant iff they are $ ext{Z}/2$-homologous and satisfy boundary or Euler number conditions.

In simply-connected 4-manifolds, closed non-orientable cobordant surfaces are concordant.

New constructions of cobordisms with prescribed boundaries and conditions for extending cobordisms.

Abstract

We show that two properly embedded compact surfaces in an orientable 4-manifold are cobordant if and only if they are $\mathbb{Z}/2$-homologous and either the 4-manifold has boundary or the surfaces have the same normal Euler number. If the 4-manifold is simply-connected and the surfaces are closed, non-orientable, and cobordant, we show that they are in fact concordant. This completes the classification of closed surfaces in simply-connected 4-manifolds up to concordance. Our methods give new constructions of cobordisms with prescribed boundaries, and completely determine when a given cobordism between the boundaries extends to a cobordism or concordance between the surfaces. We obtain our concordance results by extending Sunukjian's method of ambient surgery to the unoriented case using Pin$^-$-structures. We also discuss conditions for an arbitrary codimension 2 properly embedded submanifold to admit an unoriented spanning manifold with prescribed boundary. All results hold in both the smooth and topological categories.