Algebraic cohomotopy groups, algebraic fundamental groups and stably free modules

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.