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In mathematics, combinatorial topology was an older name for algebraic topology, dating from the time when topological invariants of spaces (for example the Betti numbers) were regarded as derived from combinatorial decompositions of spaces, such as decomposition into simplicial complexes. After the proof of the simplicial approximation theorem this approach provided rigour.

The change of name reflected the move to organise topological classes such as cycles-modulo-boundaries explicitly into abelian groups. This point of view is often attributed to Emmy Noether, and so the change of title may reflect her influence. The transition is also attributed to the work of Heinz Hopf, who was influenced by Noether, and to Leopold Vietoris and Walther Mayer, who independently defined homology.

A fairly precise date can be supplied in the internal notes of the Bourbaki group. While topology was still combinatorial in 1942, it had become algebraic by 1944. This corresponds also to the period where homological algebra and category theory were introduced for the study of topological spaces, and largely supplanted combinatorial methods.

Azriel Rosenfeld (1973) proposed digital topology for a type of image processing that can be considered as a new development of combinatorial topology. The digital forms of the Euler characteristic theorem and the Gauss–Bonnet theorem were obtained by Li Chen and Yongwu Rong. A 2D grid cell topology already appeared in the Alexandrov–Hopf book Topologie I (1935).

See also

• Hauptvermutung
• Topological combinatorics
• Topological graph theory

Notes

1. For example L'émergence de la notion de groupe d'homologie, Nicolas Basbois (PDF), (in French) note 41, explicitly names Noether as inventing homology groups.
2. Chronomaths, (in French).
3. Hirzebruch, Friedrich, "Emmy Noether and Topology" in Teicher 1999, pp. 61–63.
4. McCleary, John. "Bourbaki and Algebraic Topology" (PDF). gives documentation (translated into English from French originals).
5. Chen, Li; Rong, Yongwu (2010). "Digital topological method for computing genus and the Betti numbers". Topology and Its Applications. 157 (12): 1931–1936. doi:10.1016/j.topol.2010.04.006. MR 2646425.
6. Chen, Li; Rong, Yongwu. Linear Time Recognition Algorithms for Topological Invariants in 3D. 19th International Conference on Pattern Recognition (ICPR 2008). pp. 3254–7. arXiv:0804.1982. CiteSeerX 10.1.1.312.6573. doi:10.1109/ICPR.2008.4761192. ISBN 978-1-4244-2174-9.

References

• Alexandrov, Pavel S. (1956), Combinatorial Topology Vols. I, II, III, translated by Horace Komm, Graylock Press, MR 1643155
• Hilton, Peter (1988), "A Brief, Subjective History of Homology and Homotopy Theory in This Century", Mathematics Magazine, 60 (5), Mathematical Association of America: 282–291, doi:10.1080/0025570X.1988.11977391, JSTOR 2689545
• Teicher, Mina, ed. (1999), The Heritage of Emmy Noether, Israel Mathematical Conference Proceedings, Bar-Ilan University/American Mathematical Society/Oxford University Press, ISBN 978-0-19-851045-1, OCLC 223099225
• Novikov, Sergei P. (2001) , "Combinatorial topology", Encyclopedia of Mathematics, EMS Press

v t e Topology
Fields	General (point-set) Algebraic Combinatorial Continuum Differential Geometric low-dimensional Homology cohomology Set-theoretic Digital
Key concepts	Open set / Closed set Interior Continuity Space compact connected Hausdorff metric uniform Homotopy homotopy group fundamental group Simplicial complex CW complex Polyhedral complex Manifold Bundle (mathematics) Second-countable space Cobordism
Metrics and properties	Euler characteristic Betti number Winding number Chern number Orientability
Key results	Banach fixed-point theorem De Rham cohomology Invariance of domain Poincaré conjecture Tychonoff's theorem Urysohn's lemma
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• Algebraic topology
• Combinatorics