Algebraic cohomotopy groups, algebraic fundamental groups and stably free modules

TL;DR

This paper establishes algebraic analogs of classical topological cohomology results and demonstrates how algebraic cohomotopy groups can produce non-free stably free modules, linking topology and algebra.

Contribution

It proves algebraic versions of topological cohomology groups and connects algebraic cohomotopy to the construction of non-free stably free modules.

Findings

Algebraic cohomology groups mirror topological results for spheres.

Non-trivial algebraic cohomotopy elements yield non-free stably free modules.

Provides explicit examples linking algebraic topology and module theory.

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.