Combinatorial topology

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,^[1] and so the change of title may reflect her influence. The transition is also attributed to the work of Heinz Hopf,^[2] who was influenced by Noether, and to Leopold Vietoris and Walther Mayer, who independently defined homology.^[3]

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.^[4]

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 Euler characteristic theorem and Gauss–Bonnet theorem were obtained by Chern et al. (See digital topology.) In history, a 2D grid cell topology had appeared in Alexandrov-Hopf's book Topologie I (1935).

See also

• Topological combinatorics
• Topological graph theory

Notes

1. For example L'émergence de la notion de groupe d'homologie, Nicolas Basbois (PDF), (French) note 41, explicitly names Noether as inventing the homology group.
2. Chronomaths, (French).
3. Hirzebruch, Friedrich, "Emmy Noether and Topology" in Teicher 1999, pp. 61–63.
4. Bourbaki and Algebraic Topology by John McCleary (PDF) gives documentation (translated into English from French originals).

References

• Alexandrov, P.S.; Tr. Horace Komm (1956), Combinatorial Topology Vols. I,II,III, Graylock Press
• Hilton, Peter (1988), "A Brief, Subjective History of Homology and Homotopy Theory in This Century", Mathematics Magazine, Mathematical Association of America, 60 (5): 282–291, JSTOR 2689545
• Teicher, M. (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
• S.P. Novikov (2001), "Combinatorial topology", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4