Topological Homotopy Groups

TL;DR

This paper extends the concept of topological fundamental groups to higher homotopy groups, exploring their properties and conditions for discreteness, and argues their potential greater usefulness in topological studies.

Contribution

It introduces the notion of topological homotopy groups, investigates their properties, and establishes criteria for when their topology is discrete, expanding the understanding of topological invariants.

Findings

Topological homotopy groups can have discrete or non-discrete topologies.

Conditions for the topology of homotopy groups to be discrete are characterized.

Studying topological homotopy groups may offer more insights than fundamental groups.

Abstract

D. K. Biss (Topology and its Applications 124 (2002) 355-371) introduced the topological fundamental group and presented some interesting basic properties of the notion. In this article we intend to extend the above notion to homotopy groups and try to prove some similar basic properties of the topological homotopy groups. We also study more on the topology of the topological homotopy groups in order to find necessary and sufficient conditions for which the topology is discrete. Moreover, we show that studying topological homotopy groups may be more useful than topological fundamental groups.