Algebraic cohomotopy groups, algebraic fundamental groups and stably free modules

Abstract: In this article, we prove the algebraic counterpart of the topological results $$H^{1(S^1,} \mathbb{Z}) \cong \mathbb{Z}$$ and $H^{1(S^2,} \mathbb{Z}) \cong {0}$. We also see that a non-trivial element of the algebraic cohomotopy groups of certain rings associated with some known topological spaces provides examples of non-free stably free module of rank two over those rings.