Existence of finitely dominated CW - complexes with G 1 ( X ) = π 1 ( X ) and non - vanishing finiteness obstruction by Wolfgang Lück and

We show for a finite abelian group G and any element in the image of the Swan homomorphism sw : Z/|G|∗ −→ K̃0(ZG) that it can be realized as the finiteness obstruction of a finitely dominated connected CW -complex X with fundamental group π1(X) = G such that π1(X) is equal to the subgroup G1(X) defined by Gottlieb. This is motivated by the observation that any H-space X satisfies π1(X) = G1(X) and still the problem is open whether any finitely dominated H-space is up to homotopy finite. key words: finiteness obstruction, Swan homomorphisms, Gottlieb’s subgroup of the fundamental group. AMS-classification number: 57Q12 The purpose of this note is to prove Theorem 1 Let G be a finite abelian group and η be any element in the image of the Swan homomorphism sw : Z/|G|∗ −→ K̃0(ZG). Then there is a finitely dominated connected CW -complex X with the following properties: 1. G = π1(X); 2. Gottlieb’s subgroup G1(X) ⊂ π1(X) is equal to π1(X); 3. Wall’s finiteness obstruction õ(X) is η. Recall that a space X satisfies π1(X) = G1(X) if and only if π1(X) is abelian and for each w ∈ π1(X) the associated deck transformation on the universal covering lw : X̃ −→ X̃, which is a π1(X)-equivariant map, is π1(X)homotopic to the identity [2]. The finiteness obstruction of a finitely dominated CW -complex was introduced by Wall [10]. A survey about the finiteness obstruction and nilpotent and simple spaces is given in [7]. The Swan