Quillen Suslin theory for algebraic fundamental group

TL;DR

This paper extends Quillen Suslin theory to the algebraic fundamental group of rings, establishing a surjective homomorphism for real numbers and proposing future research problems.

Contribution

It introduces a novel approach to the algebraic fundamental group within Quillen Suslin theory and constructs a specific surjective homomorphism for real numbers.

Findings

Established a surjective homomorphism from the algebraic fundamental group of real numbers to integers.

Proposed open problems related to algebraic fundamental groups of specific rings.

Abstract

In this paper, we attempt to develop the Quillen Suslin theory for the algebraic fundamental group of a ring. We give a surjective group homomorphism from the algebraic fundamental group of the field of the real numbers to the group of integers. At the end of the paper, we also propose some problems related to the algebraic fundamental group of some particular type of rings.