Finiteness of homotopy groups related to the non-abelian tensor product

Abstract

By using finiteness related result of non-abelian tensor product we prove finiteness conditions for the homotopy groups π n (X) in terms of the number of tensors. In particular, we establish a quantitative version of the classical Blakers-Massey triad connectivity theorem. Moreover, we study others finiteness conditions and equivalence properties that arise from the non-abelian tensor square. In the end, we give applications to homotopy pushout, especially in the case of Eilenberg-MacLane spaces.

Finiteness conditions for the weak commutativity construction

“…In [2,3,4,5], the authors study the influence of the set of all tensors T ⊗ (G) = {g ⊗ h | g, h ∈ G} ⊆ ν(G) on the structure of the non-abelian tensor square G ⊗ G and related constructions. For instance, in [2], it was proved that the set of tensors {g ⊗ h | g, h ∈ G} is finite if and only if the non-abelian tensor square G ⊗ G is finite.…”