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

## TL;DR

This paper investigates conditions under which homotopy groups are finite, using properties of the non-abelian tensor product, and provides quantitative and application-based insights into classical theorems.

## Contribution

It establishes new finiteness criteria for homotopy groups based on non-abelian tensor products and extends classical theorems with quantitative versions.

## Key findings

- Finiteness conditions for homotopy groups in terms of tensors
- A quantitative version of the Blakers-Massey theorem
- Applications to homotopy pushouts and Eilenberg-MacLane spaces

## 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.