On the realizability of Gottlieb groups

A note on Rhodes and Gottlieb-Rhodes groups

“…As we noted earlier, Oprea and Strom [15] proved the n = 1 case of the following theorem. They showed that the higher dimensional version of the following theorem does not hold in ordinary Gottlieb groups.…”

“…PROOF. The n = 1 case was shown in [15,Theorem 3.1]. Let n ≥ 2 be a positive integer and Γ ∼ = ( s i=1 Z) ⊕ ( t j =1 Z m j ).…”

“…For the third question, recall the following question posed by Oprea and Strom [15]: for a finitely generated abelian group Γ and a positive integer n, is there a finite CW -complex or a compact manifold X with G n (X) ∼ = Γ ? They proved that for any finitely generated abelian group Γ , there exists a compact manifold X with G 1 (X) ∼ = Γ .…”

“…It can be seen in Lemma 4.14 that if Γ is an arbitrary abelian group and n is any positive integer, then τ n (K(Γ, n)) = Gτ n (K(Γ, n)) ∼ = Γ . Thus if there is no restrictions on spaces, then the third question about Fox-groups and Gottlieb-Fox groups is trivial like the ordinary Gottlieb groups ( [15]). In Theorems 4.15 and 4.16, we prove that for a finitely generated abelian group Γ and any positive integer n, there exists a compact manifold X which is a Gottlieb-Fox space such that τ n (X) = Gτ n (X) ∼ = Γ .…”