Homotopy classification of closed polygonal lines

TL;DR

This paper introduces fundamental topological concepts such as homotopy, degree, and fundamental group through the elementary example of closed polygonal lines in the plane, making these ideas accessible to non-specialists.

Contribution

It provides an expository overview of key topological methods applied to polygonal lines, bridging pure topology and applications in computer science.

Findings

Illustrates basic topological ideas using polygonal lines

Connects topology concepts to computer science applications

Accessible to mathematicians and students new to topology

Abstract

In this text we expose basic cases of some fundamental ideas and methods of topology. Namely, of homotopy, degree, fundamental group, covering, Whitehead invariant, etc. This is done by considering the elementary example: closed polygonal lines in a subset of the plane. Although these ideas and methods are parts of topology, they are used in other areas including computer science. This text is expository and is accessible to mathematicians not specialized in the area (and to students). The English version mostly consists of results and problems, and is followed by a more narrative Russian version having a different set of authors.