Date: 2024-05-19 | Article ID: 33600 |

The concept of a principal G-bundle over a base space M is the mathematical foundation for the concept of a gauge field, where G is the gauged group and M is space or spacetime. A trivial principal bundle M\times G→ M exists for every combination of G and M. Nontrivial principle bundles exist for some combinations of G and M but not for others. When they do exist, they may be constructed using what this article calls **patches** – trivial principal G-bundles over parts of the base space M, glued together using **transition functions**, also called **clutching functions**. This article uses that approach to derive some results about the (non)existence of nontrivial principal G-bundles when G is a compact Lie group and when the base space is an n-dimensional sphere or an n-dimensional torus, for various values of n.

**2024-05-19**(first version)

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