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.