Paths of the directed suspension

TL;DR

This paper demonstrates that the loop space of the directed suspension of a directed space is homotopy equivalent to the James construction, showing independence from the directed structure.

Contribution

It establishes a homotopy equivalence between the loop space of the directed suspension and the James construction, revealing a fundamental topological property.

Findings

Loop space of directed suspension is homotopy equivalent to James construction.

Homotopy equivalence is independent of the directed structure.

Provides new insights into the topology of directed spaces.

Abstract

We prove that the loop space of the directed suspension of a directed space is homotopy equivalent to the James construction. In particular, it does not depend on the directed structure of a given directed space.