From Wikipedia, the free encyclopedia
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] 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.
[5]
[6] A 2D
grid cell topology already appeared in the Alexandrov–Hopf book Topologie I (1935).
See also
Notes
^ 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 .
^
Chronomaths , (in French) .
^
Hirzebruch, Friedrich , "Emmy Noether and Topology" in
Teicher 1999 , pp. 61–63.
^ McCleary, John.
"Bourbaki and Algebraic Topology" (PDF) . gives documentation (translated into English from French originals).
^ 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 .
^ 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) [1994],
"Combinatorial topology" ,
Encyclopedia of Mathematics ,
EMS Press