A number theoretic classification of toroidal solenoids

TL;DR

This paper extends the classification of toroidal solenoids, using number theory and cohomology, from the 2-dimensional case to arbitrary dimensions, providing a broader understanding of their structure.

Contribution

It generalizes the classification of toroidal solenoids from 2D to n-dimensional cases using algebraic number theory and cohomology.

Findings

Classified toroidal solenoids for arbitrary dimensions using cohomology.

Connected algebraic number theory with topological classification.

Extended previous 2D classification results to higher dimensions.

Abstract

We classify toroidal solenoids defined by non-singular $n\times n$-matrices $A$ with integer coefficients by studying associated first \^Cech cohomology groups. In a previous work, we classified the groups in the case $n=2$ using generalized ideal classes in the splitting field of the characteristic polynomial of $A$. In this paper we explore the classification problem for an arbitrary $n$.