PreprintCC BY 4.0HAL
The homotopy types of directed path and trace spaces
Abstract
We construct a saturated directed space with a Hausdorff Δ-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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BibTeX
@misc{gaucher2026homotopy,
title = {{The homotopy types of directed path and trace spaces}},
author = {Philippe Gaucher},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11454},
eprint = {2609.11454},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05746183v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The homotopy types of directed path and trace spaces
AU - Philippe Gaucher
PY - 2026
DA - 2026/09/10
PB - HAL
DO - 10.48550/arxiv.2609.11454
UR - https://hal.science/hal-05746183v1
AB - We construct a saturated directed space with a Hausdorff Δ-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.
ER -