Topological classification of oriented cycles of linear mappings

TL;DR

This paper addresses the classification of oriented cycles of linear mappings over real or complex fields, reducing the problem to classifying linear operators up to homeomorphism, building on prior foundational work.

Contribution

It introduces a reduction of the classification problem of oriented cycles to the well-studied problem of classifying linear operators up to homeomorphism.

Findings

Reduction of cycle classification to operator classification

Connection to Kuiper and Robbin's work on linear operators

Framework for classifying cycles in terms of known operator results

Abstract

We consider the problem of classifying oriented cycles of linear mappings $F^p\to F^q\to\dots\to F^r\to F^p$ over a field $F$ of complex or real numbers up to homeomorphisms in the spaces $F^p,F^q,\dots,F^r$. We reduce it to the problem of classifying linear operators $F^n\to F^n$ up to homeomorphism in $F^n$, which was studied by N.H. Kuiper and J.W. Robbin [Invent. Math. 19 (2) (1973) 83-106] and by other authors.