Book contents
- Frontmatter
- Contents
- Preface
- Introduction
- 1 Traces and Euler characteristics
- 2 Groups of virtually finite dimension
- 3 Free abelianised extensions of finite groups
- 4 Arithmetic groups
- 5 Topological methods in group theory
- 6 An example of a finite presented solvable group
- 7 SL3(Fq[t]) is not finitely presentable
- 8 Two-dimensional Poincaré duality groups and pairs
- 9 Metabelian quotients of finitely presented soluble groups are finitely presented
- 10 Soluble groups with coherent group rings
- 11 Cohomological aspects of 2-graphs. II
- 12 Recognizing free factors
- 13 Trees of homotopy types of ( m)-complexes
- 14 Geometric structure of surface mapping class groups
- 15 Cohomology theory of aspherical groups and of small cancellation groups
- 16 Finite groups of deficiency zero
- 17 Äquivalenzklassen von Gruppenbeschreibungen, Identitäten und einfacher Homotopietyp in niederen Dimensionen
- 18 Two-dimensional complexes with torsion values not realizable by self-equivalences
- 19 Applications of Nielsen's reduction method to the solution of combinatorial problems in group theory: a survey
- 20 Chevalley groups over polynomial rings
- List of problems
18 - Two-dimensional complexes with torsion values not realizable by self-equivalences
Published online by Cambridge University Press: 05 April 2013
- Frontmatter
- Contents
- Preface
- Introduction
- 1 Traces and Euler characteristics
- 2 Groups of virtually finite dimension
- 3 Free abelianised extensions of finite groups
- 4 Arithmetic groups
- 5 Topological methods in group theory
- 6 An example of a finite presented solvable group
- 7 SL3(Fq[t]) is not finitely presentable
- 8 Two-dimensional Poincaré duality groups and pairs
- 9 Metabelian quotients of finitely presented soluble groups are finitely presented
- 10 Soluble groups with coherent group rings
- 11 Cohomological aspects of 2-graphs. II
- 12 Recognizing free factors
- 13 Trees of homotopy types of ( m)-complexes
- 14 Geometric structure of surface mapping class groups
- 15 Cohomology theory of aspherical groups and of small cancellation groups
- 16 Finite groups of deficiency zero
- 17 Äquivalenzklassen von Gruppenbeschreibungen, Identitäten und einfacher Homotopietyp in niederen Dimensionen
- 18 Two-dimensional complexes with torsion values not realizable by self-equivalences
- 19 Applications of Nielsen's reduction method to the solution of combinatorial problems in group theory: a survey
- 20 Chevalley groups over polynomial rings
- List of problems
Summary
STATEMENT OF PROBLEM AND DISCUSSION OF RESULT
The study of the group ϵ(K2) of self-equivalences of a twodimensional complex K2 has so far led only to cases, where all values of the Whitehead group can be realized as τ(f), [f] ϵ ε(K2), see Cockroft and Moss [2], Dyer and Sieradski [5] and Olum [8], It is the aim of the present paper to show by examples that this is not true in general; there exist nonrealizable torsion values even for finitely generated fundamental groups:
Theorem 1. The standard complex K2of the presentation {a, b∣ bP, a b a−1b−1 } of π=ℤ × ℤp, P prime, has nonrealizable torsion values if and only if the class number, h(p), of the p-th cyclotomic field is different from 1.
This result has consequences for the problem, which torsion values lie in Wh*(π), and the still unsolved question (see Cohen [3], p. 81 and problem D6 of this volume), whether homotopy type equals simple-homotopy type for all finite 2-complexes:
Theorem 2. If τ0is nonrealizable with respect to ε(K2), but-τ ϵWh*(π1(K2)), then the corresponding extension L2 ⊃ K2gives rise to complexes L and K with the same homotopy type but different simple-homotopy types.
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- Homological Group Theory , pp. 327 - 338Publisher: Cambridge University PressPrint publication year: 1979
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