A Whitehead algorithm for surface groups
type: Item
instance of: scholarly article
title: A Whitehead algorithm for surface groups
zbMATH Open document ID: 0973.20024
DOI: 10.1016/S0040-9383(99)00027-0
author: Gilbert Levitt
author: Karen Vogtmann
published in: Topology
publication date: 2001-06-24T00:00:00Z
review text: A problem about a group and its automorphisms: Given finite sets \(\{a_1,\dots,a_n\}\) and \(\{b_1,\dots,b_n\}\) of elements of a group \(G\), is there an automorphism of \(G\) taking \(a_i\) to \(b_i\) for all \(i\)? If so, find one. In 1936, J.~H.~C. Whitehead found an elegant algorithm which solves this problem for a finitely generated free group. In 1984, Collins and Zieschang extended Whitehead's methods to free products of finitely many freely indecomposable groups, assuming that the above problem, which will be called the Whitehead problem, can be solved in each factor. In this paper, the authors give an algorithm to solve the Whitehead problem for the fundamental group of a closed surface \(\Sigma\) and a surface with boundary. If one represents the elements \(a_i\) and \(b_i\) in \(\pi_1(\Sigma,p)\) by loops, the problem then translates into the following problem about closed curves on a surface: Is there a diffeomorphism of \(\Sigma\) fixing \(p\) which takes each \(\alpha_i\) to a loop homotopic to \(\beta_i\)? If so, find one. The solution to this geometric problem relies in part on work of Hass and Scott on straightening curves on surfaces.
Mathematics Subject Classification ID: 20F34
Mathematics Subject Classification ID: 57M07
Mathematics Subject Classification ID: 20E36
Mathematics Subject Classification ID: 20F10
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 57M50
zbMATH DE Number: 1526239
zbMATH Keywords: Whitehead problem
zbMATH Keywords: fundamental groups of closed surfaces
zbMATH Keywords: automorphisms
zbMATH Keywords: free products
reviewed by: V. V. Chueshev
MaRDI profile type: Publication
Wikidata QID: Q127878358
full work available at URL: s0040-9383(99)00027-0
OpenAlex ID: W2082663834
description: scientific article; zbMATH DE number 1526239