The fundamental group of reduced suspensions

TL;DR

This paper classifies pointed spaces based on the fundamental group of their reduced suspension, distinguishing between totally path disconnected and horseshoe types, and shows the fundamental group as a complete invariant for a certain topological equivalence.

Contribution

It introduces a topological classification of pointed spaces via the fundamental group of their reduced suspension and characterizes these spaces in terms of fundamental group invariants.

Findings

Spaces are classified into two types: totally path disconnected and horseshoe.

Fundamental group serves as a complete invariant for a weaker topological equivalence among totally path disconnected spaces.

A topological characterization of the two space types is provided based on fundamental groups.

Abstract

We classify pointed spaces according to the first fundamental group of their reduced suspension. A pointed space is either of so-called totally path disconnected type or of horseshoe type. These two camps are defined topologically but a characterization is given in terms of fundamental groups. Among totally path disconnected spaces the fundamental group is shown to be a complete invariant for a notion of topological equivalence weaker than that of homeomorphism.