#### 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 complexes.

##### Keywords
minimal genus, two-complex, undecidability
Primary: 57R95
##### 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