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

Using finiteness-related results for non-abelian tensor products, 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 other finiteness conditions and equivalence properties that arise from the non-abelian tensor square. Finally, we give applications to homotopy pushouts, especially in the case of Eilenberg–MacLane spaces.