The simple loop conjecture
type: Item
instance of: scholarly article
title: The simple loop conjecture
zbMATH Open document ID: 0556.57007
author: David Gabai
published in: Journal of Differential Geometry
publication date: 1985-01-01T00:00:00Z
review text: The following is the main result of this paper. Theorem. If \(f: S\to T\) is a map of closed connected surfaces such that \(f_*: \pi_ 1(S)\to \pi_ 1(T)\) is not injective, then there exists a noncontractible simple closed curve \(\alpha\) \(\subset S\) such that \(f| \alpha\) is homotopically trivial. A proof is given of the following known but unpublished result. Theorem. If \(f: S\to T\) is a map of bounded connected compact surfaces such that \(f_*: \pi_ 1(S)\to \pi_ 1(T)\) is not injective, then there exists an essential simple arc \(\alpha\) \(\subset S\) and a map g homotopic to f such that g(\(\alpha)\) is a boundary parallel arc.
Mathematics Subject Classification ID: 57N05
Mathematics Subject Classification ID: 57M12
Mathematics Subject Classification ID: 57M35
Mathematics Subject Classification ID: 57M05
zbMATH DE Number: 3886760
zbMATH Keywords: simple loops in the kernel of maps of fundamental groups
zbMATH Keywords: branched cover
zbMATH Keywords: map of closed connected surfaces
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214439470
OpenAlex ID: W1571267456
Wikidata QID: Q115187304
DOI: 10.4310/JDG/1214439470
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description: scientific article; zbMATH DE number 3886760