Concordance of spheres in 4-manifolds with an immersed dual sphere

2022/11/14 by Michael J. Klug, Klug, Michael, Maggie Miller +1 · 1 citation
Computer Science · Mathematics · #57K45 #57K99 #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2211.07177

openalex publication_date 2022/11/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let S₀ and S₁ be two homotopic, oriented 2-spheres embedded in an orientable 4-manifold X. 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 S₀,S₁, they are defined in terms of the self-intersection set of a regular homotopy from S₀ to S₁. When S₀ has an immersed dual sphere, we see that under some mild topological conditions on X, 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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• Simple spheres in simply-connected 4-manifolds

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