Topologically isotopic and smoothly inequivalent 2–spheres in simply connected 4–manifolds whose complement has a prescribed fundamental group

Torres, Rafael

doi:10.2140/agt.2024.24.2351

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Topologically isotopic and smoothly inequivalent $2$–spheres in simply connected $4$–manifolds whose complement has a prescribed fundamental group

Rafael Torres

Algebraic & Geometric Topology 24 (2024) 2351–2365

DOI: 10.2140/agt.2024.24.2351

Abstract

We describe a procedure to construct infinite sets of pairwise smoothly inequivalent $2$ –spheres in simply connected $4$ –manifolds, which are topologically isotopic and whose complement has a prescribed fundamental group that satisfies some conditions. This class of groups include cyclic groups and the binary icosahedral group. These are the first known examples of such exotic embeddings of $2$ –spheres in $4$ –manifolds. Examples of locally flat embedded $2$ –spheres in a nonsmoothable $4$ –manifold whose complements are homotopy equivalent to smoothly embedded ones are also given.

Keywords

knotted surfaces, 4–manifolds

Mathematical Subject Classification

Primary: 57K45, 57R55

Secondary: 57R40, 57R52

References

Publication

Received: 1 December 2022

Revised: 27 March 2023

Accepted: 28 April 2023

Published: 16 July 2024

Authors

Rafael Torres
Scuola Internazionale Superiori di Studi Avanzati (SISSA) Trieste Italy

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Volume 24, issue 4 (2024)