Concordance of spheres in 4-manifolds with an immersed dual sphere
Michael Klug, Maggie Miller

TL;DR
This paper investigates conditions under which two homotopic 2-spheres in a 4-manifold are concordant, showing that certain invariants are complete obstructions when an immersed dual sphere exists, adapting Stong's methods.
Contribution
It demonstrates that the Freedman–Quinn and Stong invariants form a complete set of concordance obstructions for spheres with an immersed dual sphere in specific 4-manifolds, extending Stong's techniques.
Findings
Invariants fq and stong are complete obstructions under certain conditions.
The work adapts Stong's methods to sphere concordances.
Conditions on the 4-manifold ensure invariants fully classify concordance.
Abstract
Let and be two homotopic, oriented 2-spheres embedded in an orientable 4-manifold . After discussing several operations for modifying an immersion of a 3-manifold into a 5-manifold, we discuss the Freedman--Quinn (fq) and Stong (stong) concordance obstructions. When these are defined for the pair , they are defined in terms of the self-intersection set of a regular homotopy from to . When has an immersed dual sphere, we see that under some mild topological conditions on , the invariants fq and stong are a complete set of concordance obstructions. This work is an adaption of the methods of Richard Stong to the context of concordances of 2-spheres.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Topological and Geometric Data Analysis · Geometric and Algebraic Topology
