nLab computational topology

Overview

Computational topology is a relatively new area of study. It emerged in response to topological problems that arise in computer graphics, robotics and planning. There are interactions with dynamical systems, and computational geometry.

Subareas

• applied topology

• computational homology?

• persistent homology

• topological data analysis

• topological dynamics

• discrete Morse theory

Related areas include

• computational homological algebra

See also:

• constructive set theory

• constructive analysis

References

General

Textbook accounts:

• Herbert Edelsbrunner, John L. Harer, Computational Topology, An Introduction, American Mathematical Society (2010) [ISBN:978-0-8218-4925-5, pdf]

• Graham Ellis, An Invitation to Computational Homotopy, Oxford University Press (2019) [ISBN:9780198832973, MAA review]

See also:

• sources in computational topology

Computational algebraic topology

• Rolf Schön: Effective Algebraic Topology, Memoirs of the AMS 451, AMS (1991) [ams:memo-92-451]

Constructive$\;$homology groups and homotopy groups:

• Julio Rubio, Francis Sergeraert, Constructive Algebraic Topology, Bulletin des Sciences Mathématiques

  126 5 (2002) 389-412 [arXiv:math/0111243, doi:10.1016/S0007-4497(02)01119-3]

• Kenzo (computer system for computational topology)

Computational Homotopy groups

Discussion of (equivariant) homotopies and homotopy groups in computational topology:

• Martin Čadek, Marek Krčál, Jiří Matoušek, Lukáš Vokřínek, Uli Wagner, Polynomial-time computation of homotopy groups and Postnikov systems in fixed dimension, SIAM J. Comput., 43(5), 1728–1780 (arXiv:1211.3093, doi:10.1137/120899029)

• Lukáš Vokřínek, Computing the abelian heap of unpointed stable homotopy classes of maps, Archivum Mathematicum, Volume 49 (2013), No. 5 · (arXiv:1312.2474, doi:10.5817/AM2013-5-359)

• Marek Filakovský, Lukáš Vokřínek, Are two given maps homotopic? An algorithmic viewpoint, Found Comput Math (2019) (arXiv:1312.2337, doi:10.1007/s10208-019-09419-x)

• Marek Filakovsky, Peter Franek, Uli Wagner, Stephan Zhechev, Computing simplicial representatives of homotopy group elements, J Appl. and Comput. Topology (2018) 2: 177. (arXiv:1706.00380)

and on the extension-problem:

• Martin Čadek, Marek Krčál, Jiří Matoušek, Lukáš Vokřínek, Uli Wagner, Extendability of continuous maps is undecidable, Discrete Comput Geom (2014) 51: 24 (arXiv:1302.2370, doi:10.1007/s00454-013-9551-8)

Computational Cohomotopy sets

Discussion of Cohomotopy-sets in computational topology:

• Martin Čadek, Marek Krčál, Jiří Matoušek, Francis Sergeraert, Lukáš Vokřínek, Uli Wagner, Computing all maps into a sphere, Journal of the ACM, Volume 61 Issue 3, May 2014 Article No. 1 (arxiv:1105.6257)

• Lukáš Vokřínek, Decidability of the extension problem for maps into odd-dimensional spheres, Discrete Comput Geom (2017) 57: 1 (arXiv:1401.3758)