Combinatorial topology

Source: Wikipedia, the free encyclopedia.

In

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] 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.

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.[5][6] A 2D grid cell topology
already appeared in the Alexandrov–Hopf book Topologie I (1935).

See also

Notes

  1. 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. MR 2646425
    .
  6. ISBN 978-1-4244-2174-9
    .

References
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