Surfaces in 4-manifolds: Concordance, Isotopy, and Surgery

TL;DR

This paper explores the relationships between surfaces in 4-manifolds, demonstrating concordance, conditions for isotopy, and equivalence of surgeries, thereby advancing understanding of 4-manifold topology.

Contribution

It establishes new results on concordance, isotopy, and surgery equivalences for surfaces in 4-manifolds, linking these concepts under specific conditions.

Findings

Surfaces of same genus and homology class are concordant in simply connected 4-manifolds.

Topological isotopy often occurs when complements have cyclic fundamental group.

Surgery on 0-concordant surfaces is equivalent, showing a new relation between surgery and concordance.

Abstract

In this paper we will show that two surfaces of the same genus and homology class in a simply connected 4-manifold are concordant. We will show they are often topologically isotopic when their complements have cyclic fundamental group. Finally, we will show that if they are 0-concordant, then surgery on one is equivalent to surgery on the other.