Nonminimal roots in homotopy trees

Micheal Neal Dyer

Vol. 80 (1979), No. 2, 371–380

DOI: 10.2140/pjm.1979.80.371

Abstract

Let π be a finite group which does not satisfy the Eichler condition and let M be a π-module. A π-module M′ is a noncancellation example of M if M ⊕ (Zπ)²≅M′⊕ (Zπ)² but M≇M′. This note classifies the set 𝒩𝒞_M(π) of isomorphism classes of noncancellation examples for M = Z ⊕ Zπ, where Z is the trivial π-module, M = A(π), the augmentation ideal, and M = Zπ∕(N), where (N) is the ideal generated by the norm element N = ∑ _x∈πx. It is shown that these noncancellation examples yield nonminimal roots of the homotopy tree HT(π,m) of (π,m)-complexes.

Mathematical Subject Classification 2000

Primary: 57M20

Secondary: 16A26

Milestones

Received: 13 November 1975

Revised: 1 February 1978

Published: 1 February 1979