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Nielsen Equivalence and Simple‐Homotopy Type
Author(s) -
Lustig Martin
Publication year - 1991
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-62.3.537
Subject(s) - mathematics , homotopy , equivalence (formal languages) , simple (philosophy) , homotopy group , vertex (graph theory) , whitehead theorem , n connected , combinatorics , pure mathematics , homotopy sphere , regular homotopy , type (biology) , ecology , philosophy , epistemology , biology , graph
Nielsen equivalence and simple‐homotopy equivalence are interpreted as analogues in dimensions 1 and 2 respectively. This yields the following results. § 1. A new method for distinguishing Nielsen equivalence classes of generating systems in a group G is presented. As an application, inequivalent generating systems are exhibited in metabelian groups and in cocompact Fuchsian groups (extending the results of [ 12 ]). § 2. The techniques developed in § 1 are applied to the simple‐homotopy theory of 2‐complexes, and examples of pairs of 2‐complexes K , L are constructed, which are homotopy equivalent, but not simple‐homotopy equivalent. The simplest consists of the one‐vertex 2‐complexes for the presentations K: (x,y,s|y 3 ,yx 10 y −1 x −5 ,[x 7 ,z]), L: (x,y,z|y 3 ,yx 10 y −1 x −5 ,x 14 zx 14 z −1 x −7 zx −21 z −1 ) .