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

Kastenholz, Thorben; Pedron, Mark

doi:10.2140/agt.2022.22.1375

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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

DOI: 10.2140/agt.2022.22.1375

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.

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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

Volume 22, issue 3 (2022)