Volume 14, Issue 2 p. 560-586
Research Article

Concordance of surfaces in 4-manifolds and the Freedman–Quinn invariant

Michael R. Klug,

Department of Mathematics, University of California, Berkeley, 970 Evans Hall, Berkeley, CA, 94720 USA

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Maggie Miller,

Corresponding Author

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 02139 USA

maggiehm@mit.edu

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First published: 10 May 2021

MK was supported by the Max Planck Institute for Mathematics (MPIM) in Bonn, Germany during the time of this project. MM was supported by MPIM during part of this project, as well as NSF Grants DGE-1656466 and DMS-2001675.

Abstract

We prove a concordance version of the 4-dimensional light bulb theorem for π 1 -negligible compact orientable surfaces, where there is a framed but not necessarily embedded dual sphere. That is, we show that if F 0 and F 1 are such surfaces in a 4-manifold X that are homotopic and there exists an immersed framed 2-sphere G in X intersecting F 0 geometrically once, then F 0 and F 1 are concordant if and only if their Freedman–Quinn invariant fq vanishes. The proof of the main result involves computing fq in terms of intersections in the universal covering space and then applying work of Sunukjian in the simply-connected case. This paper relies extensively on colour figures. Some references to colour may not be meaningful in the printed version, and we refer the reader to the online version which includes the colour figures.