Abstract
Here, we introduce a new definition of regular point for piecewise-linear (PL) functions on combinatorial (PL triangulated) manifolds. This definition is given in terms of the restriction of the function to the link of the point. We show that our definition of regularity is distinct from other definitions that exist in the combinatorial topology literature. Next, we stratify the Jacobi set/critical locus of such a map as a poset stratified space. As an application, we consider the Reeb space of a PL function, stratify the Reeb space as well as the target of the function, and show that the Stein factorization is a map of stratified spaces.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
We thank the referee for some specific suggested language around this motivation.
References
Ayala, D., Francis, J., Tanaka, H.L.: Local structures on stratified spaces. Adv. Math. 307, 903–1028 (2017)
Bauer, B., Ulrich, G.X., Wang, Y., Measuring distance between reeb graphs. In Proceedings of the thirtieth annual symposium on Computational geometry, pages 464–473, (2014)
Biasotti, S., Giorgi, D., Spagnuolo, M., Falcidieno, B.: Reeb graphs for shape analysis and applications. Theoretical Comput. Sci. 392(1–3), 5–22 (2008)
Björner, A.: Posets, regular cw complexes and bruhat order. Euro. J. Combinatorics 5(1), 7–16 (1984)
Cole-McLaughlin, K., Edelsbrunner, H., Harer, J., Natarajan, V., Pascucci, V.: Loops in reeb graphs of 2-manifolds. Discrete Comput. Geometry 32(2), 231–244 (2004)
De Silva, V., Munch, E., Patel, A.: Categorified reeb graphs. Discrete Comput. Geometry 55(4), 854–906 (2016)
Dey, T.K., Fan, F., Wang, Y.: An efficient computation of handle and tunnel loops via reeb graphs. ACM Trans. Graphics (TOG) 32(4), 1–10 (2013)
Doraiswamy, H., Natarajan, V.: Efficient algorithms for computing reeb graphs. Comput. Geometry 42(6–7), 606–616 (2009)
Herbert, E., John, H., Patel, A.K., Reeb spaces of piecewise linear mappings. In Proceedings of the twenty-fourth annual symposium on Computational geometry, pages 242–250, (2008)
Friedman, G.: Singular intersection homology. New Mathematical Monographs, vol. 33. Cambridge University Press, Cambridge (2020)
Hersh, P., Kenyon, R.: Shellability of face posets of electrical networks and the cw poset property. Adv. Appl. Math. 127, 102178 (2021)
Lurie, J.: Higher algebra. available at Author’s Homepage, (2022)
Nicolaescu, L.: An invitation to Morse theory, 2nd edn. Universitext. Springer, New York (2011)
Nocera, G., Volpe, M.,: Whitney stratifications are conically smooth. arXiv preprint arXiv:2105.09243, (2021)
Reeb, G.: Sur les points singuliers d’une forme de pfaff completement integrable ou d’une fonction numerique [on the singular points of a completely integrable pfaff form or of a numerical function]. Comptes Rendus Acad. Sci. Paris 222, 847–849 (1946)
Rourke, C.P., Sanderson, B.J.: Introduction to piecewise-linear topology. Springer-Verlag, New York-Heidelberg, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 69 (1972)
Stanley, R.P.: f-vectors and h-vectors of simplicial posets. J. Pure Appl. Algebra 71(2–3), 319–331 (1991)
Thurston, W.P.: Three-dimensional geometry and topology. Vol. 1, volume 35 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, Edited by Silvio Levy (1997)
Trotman, D.: Stratification theory. In:Handbook of geometry and topology of singularities. I, pages 243–273. Springer, Cham, [2020] (2020)
Acknowledgements
AS is supported by the National Science Foundation under NIH/NSF DMS 1664858. RG is supported by the Simons Foundation under Travel Support/Collaboration 9966728. The authors thank David Ayala for discussion and shared insight. Many thanks also go to the anonymous referee for their thorough and incredibly helpful suggestions.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Grady, R., Schenfisch, A. Regularity via links and Stein factorization. Beitr Algebra Geom (2023). https://doi.org/10.1007/s13366-023-00713-y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13366-023-00713-y