# Homotopy and Commutativity Principle

## TL;DR

This paper proves a homotopy and commutativity principle for classical groups and automorphism groups of modules, leveraging a local-global principle to establish abelian or solvable quotient structures.

## Contribution

It extends the homotopy and commutativity principle to symplectic, orthogonal, and automorphism groups using a non-symmetric local-global approach.

## Key findings

- Symplectic quotients are abelian.
- Orthogonal quotients are solvable of length at most two.
- Orthogonal quotients are abelian over regular local rings containing a field.

## Abstract

In this article, we prove commutativity principal for linear, symplectic and transvection groups. This principle is a consequence of Quillen-Suslin local global principle and using a non-symmetric application of it as done by A. Bak. The existence of a Local-Global Principle enables us to prove similar results in various groups. We restrict ourselves to the classical symplectic, orthogonal groups (and their relative versions); and to the automorphism groups of a projective module (with a unimodular element), a symplectic module (with ahyperbolic summand), and an orthogonal module (with a hyperbolic symmand). We could show that the symplectic quotients were abelian, but we could only establish that the orthogonal quotients are solvable of length atmost two. We do believe that the orthogonal quotient groups are also abelian; and prove this when the base ring is a regular local ring containing a field.