Exotic relation modules and homotopy types for certain 1–relator groups

Abstract

Using stably free non-free relation modules we construct an infinite collection of 2-dimensional homotopy types, each of Euler-characteristic one and with trefoil fundamental group. This provides an affirmative answer to a question asked by Berridge and Dunwoody [1]. We also give new examples of exotic relation modules. We show that the relation module associated with the generating set {x, y 4 } for the Baumslag-Solitar group x, y | xy 2 x −1 = y 3 is stably free non-free of rank one. 57M20; 57M05

On the $K$-theory and homotopy theory of the Klein bottle group

“…It was shown in [8] that for groups that are fundamental groups of 2-dimensional aspherical complexes, relation modules can always be realized as second homotopy modules π 2 (X) for some (G, 2)-complex X. In particular, if the stably free G-modules K n constructed earlier for the Klein bottle group G could be shown to be isomorphic to relation modules, then all the algebraic 2-complexes X n of Theorem 3.1 could be geometrically realized.…”

“…Stably free non-free G-modules often arise as relation modules (see Dunwoody [5], Harlander and Jensen [8], Lustig [15]). It was shown in [8] that for groups that are fundamental groups of 2-dimensional aspherical complexes, relation modules can always be realized as second homotopy modules π 2 (X) for some (G, 2)-complex X.…”

On the Borsuk conjecture concerning homotopy domination

“…[2,10]). For infinitely many i, there are 2-dimensional CW -complexes K i of distinct homotopy type, with [10]. Note that the commutator subgroup [G, G] of G is isomorphic to F 2 , i.e.…”

Examples of exotic free 2–complexes and stably free nonfree modules for quaternion groups

“…When P is given by Example 4.2 or Example 4.3, for the ‚ of (2-9), choose h D 1, the factorˆ.e; f; n/ according to Corollary 1.7, the integer c and ‰.e; f; n/ as in (2)(3)(4)(5)(6)(7). In all of these cases P =NP is not singly generated.…”

“…If P Š ZG k whenever this condition is met, all stably free modules are free and ZG is said to have the cancellation property. Berridge-Dunwoody [1], Lewin [12] and also Harlander-Jensen [6] give examples of stably free nonfree modules for certain infinite groups. For most finite groups G , no such examples exist.…”