Wrap Groups of Non-Archimedean Fiber Bundles

TL;DR

This paper introduces and investigates the structure of wrap groups associated with fiber bundles over non-Archimedean fields and Cayley-Dickson algebras, revealing their infinite-dimensional and totally disconnected nature.

Contribution

It extends the concept of wrap groups to non-Archimedean and Cayley-Dickson algebra contexts, analyzing their smoothness and structural properties.

Findings

Wrap groups are generally infinite dimensional over the base field.

They are totally disconnected groups with non-Archimedean smoothness.

Skew products of wrap groups are also studied.

Abstract

Fiber bundles over infinite fields with non-trivial ultra-norms are considered. For them geometric wrap groups are defined and investigated. Besides fields also Cayley-Dickson algebras over fields of characteristic not equal to two are taken into account. For fibers over them wrap groups are introduced and their structure is investigated. Different classes of smoothness for wrap groups are used. It is demonstrated that generally such groups are infinite dimensional over the corresponding field and totally disconnected groups. That is, they are continuous or differentiable non-archimedean differentiable uniform spaces and the composition $(f,g)\mapsto f^{-1}g$ is continuous or differentiable depending on a class of smoothness of groups. Skew products of wrap groups are studied as well.