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 ${\pi }_n(X)$ 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.