Abstract
While collapsibility of CW complexes dates back to the 1930s, collapsibility of directed Euclidean cubical complexes has not been well studied to date. The classical definition of collapsibility involves certain conditions on pairs of cells of the complex. The direction of the space can be taken into account by requiring that the past links of vertices remain homotopy equivalent after collapsing. We call this type of collapse a link-preserving directed collapse. In the undirected setting, pairs of cells are removed that create a deformation retract. In the directed setting, topological properties—in particular, properties of spaces of directed paths—are not always preserved. In this paper, we give computationally simple conditions for preserving the topology of past links. Furthermore, we give conditions for when link-preserving directed collapses preserve the contractability and connectedness of spaces of directed paths. Throughout, we provide illustrative examples.
| Originalsprog | Engelsk |
|---|---|
| Titel | Research in Computational Topology 2 |
| Redaktører | Ellen Gasparovic, Vanessa Robins, Katharine Turner |
| Antal sider | 23 |
| Forlag | Springer |
| Publikationsdato | 2022 |
| Sider | 167-189 |
| ISBN (Trykt) | 978-3-030-95518-2 |
| ISBN (Elektronisk) | 978-3-030-95519-9 |
| DOI | |
| Status | Udgivet - 2022 |
| Navn | Association for Women in Mathematics Series |
|---|---|
| Vol/bind | 30 |
| ISSN | 2364-5733 |
Bibliografisk note
Funding Information:This research is a product of one of the working groups at the Women in Topology (WIT) workshop at MSRI in November 2017. This workshop was organized in partnership with MSRI and the Clay Mathematics Institute, and was partially supported by an AWM ADVANCE grant (NSF-HRD 1500481). This material is based upon work supported by the US National Science Foundation under grant No. DGE 1649608 (Belton) and DMS 1664858 (Fasy), as well as the Swiss National Science Foundation under grant No. 200021-172636 (Ebli). We thank the Computational Topology and Geometry (CompTaG) group at Montana State University for giving helpful feedback on drafts of this work.
Funding Information:
This material is based upon work supported by the US National Science Foundation under grant No. DGE 1649608 (Belton) and DMS 1664858 (Fasy), as well as the Swiss National Science Foundation under grant No. 200021-172636 (Ebli).
Funding Information:
Acknowledgments This research is a product of one of the working groups at the Women in Topology (WIT) workshop at MSRI in November 2017. This workshop was organized in partnership with MSRI and the Clay Mathematics Institute, and was partially supported by an AWM ADVANCE grant (NSF-HRD 1500481).
Publisher Copyright:
© 2022, The Author(s) and the Association for Women in Mathematics.