Twisted Homotopy: A Group Theoretic Approach

TL;DR

This paper introduces a novel group-theoretic approach to twisted homotopy, enhancing the fundamental group with continuous SO(2) considerations to reveal self-interactions among loops relevant in quantum mechanics.

Contribution

It presents a new group-theoretic framework for twisted homotopy by extending the fundamental group to SO(2), offering a different perspective from previous physical and cohomological approaches.

Findings

The fundamental group is extended to SO(2) via a parameter change.

This extension induces self-interactions among loops.

The angle parameter acts as a self-coupling constant.

Abstract

After summarising the physical approach leading to twisted homotopy and after developing the cohomological approach further with respect to our previous work we propose a third alternative approach to twisted homotopy based on group theoretic considerations. In this approach the fundamental group $\Pi (m) $ isomorphic to Z which describes homotopic loops on the punctured plane$ R^2/(0) $ is enhanced in a special way to the continuous SO(2) group . This is performed by letting the parameter of the group $ m \rightarrow \lambda $ while keeping its generator unchanged .It is shown that such non-trivial procedure has the effect of introducing well defined self-interactions among loops which are at the basis of twisted homotopy where the angle $ \lambda $ plays the role of the self coupling constant. KEYWORDS: Homotopy, Group Theory, Quantum Mechanics MSC:55Q35; PACS:02.20.Fh ; 03.65.Fd