Volume 22, issue 3 (2022)

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The minimal genus of homology classes in a finite 2–complex

Thorben Kastenholz and Mark Pedron

Algebraic & Geometric Topology 22 (2022) 1375–1415
Abstract

We study surface representatives of homology classes of finite complexes which minimize certain complexity measures, including their genus and Euler characteristic. Our main result is that, up to surgery at nullhomotopic curves, minimizers are homotopic to cellwise coverings of the 2–skeleton. From this we conclude that the minimizing problem is in general algorithmically undecidable, but can be solved for 2–dimensional  CAT(1) complexes.

Keywords
minimal genus, two-complex, undecidability
Mathematical Subject Classification
Primary: 57R95
References
Publication
Received: 19 October 2020
Revised: 3 March 2021
Accepted: 29 March 2021
Published: 25 August 2022
Authors
Thorben Kastenholz
University of Göttingen
Göttingen
Germany
Mark Pedron
Bonn
Germany