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arXiv · 2609.11454

The homotopy types of directed path and trace spaces

Philippe Gaucher
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Abstract

We construct a saturated directed space with a Hausdorff $\Delta$-generated underlying space and two distinct points such that the trace space between them is homeomorphic to a square, whereas the directed path space has a nontrivial fundamental group. In particular, the canonical quotient map is not a weak homotopy equivalence. The same conclusion holds for regular directed paths modulo increasing homeomorphisms.

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Philippe Gaucher. 2026-09-10. The homotopy types of directed path and trace spaces. https://arxiv.org/abs/2609.11454

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