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

Abstract

Using finiteness-related results for non-abelian tensor products, 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 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.

Boundedly finite conjugacy classes of tensors

“…The study of the non-abelian tensor square of groups from a group theoretic point of view was initiated by R. Brown, D. L. Johnson and E. F. Robertson [5]. For a deeper discussion of the non-abelian tensor square and related constructions we refer the reader to [10,11] (see also [4]).…”

“…It is well-known that the set of all tensors T ⊗ (G) affects the structure of the non-abelian tensor square G ⊗ G, and of related constructions (see [1,2,3,4]). For instance, in [1] and in [2], it was proved that if the set T ⊗ (G) is finite (resp.…”