×

Topology becomes algebraic with Vietoris and Noether. (English) Zbl 0577.01030

The author points out that homology groups were formally introduced simultaneously by Emmy Noether and Leopold Vietoris in 1926 and produces evidence that the Göttingen and Vienna schools were independent in this achievement. He also quotes a letter from Vietoris in which the latter writes ”Without doubt H. Poincaré and his contemporaries knew that the Betti numbers and torsion coefficients were invariants of groups.... Then one worked with the numerical invariants rather than with the invariant group. That was a matter of taste.” The same position is taken by L. Vietoris in his Encyclopaedie article [Tieze-Vietoris, Beziehungen zwischen den verschiedenen Zweigen der Topologie, Enc. Math. Wiss. Bd. III/1, p. 224 (1929)].
Reviewer: H.Guggenheimer

MSC:

01A60 History of mathematics in the 20th century
55-03 History of algebraic topology

Biographic References:

Noether, Emmy; Vietoris, L.
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Dieudonné, J., Emmy Noether and Algebraic Topology, J. Pure Appl. Algebra, 31, 5-6 (1984) · Zbl 0545.55001
[2] Hopf, H., A new proof of the Lefschetz formula on invariant points, (Proc. Nat. Acad. Sci. USA, 14 (1928)), 149-153
[3] Hopf, H., Über die algebraische Anzahl von Fixpunkten, Math. Z., 29, 493-524 (1929)
[4] Hopf, H., Eine Verallgemeinerung der Euler-Poincaréschen Formel, Nachr. Ges. der Wiss. zu Göttingen, 127-136 (1928)
[5] Lefschetz, S., Intersection and transformation of complexes and manifolds, Trans. Am. Math. Soc., 28, 1-49 (1926)
[6] Mayer, W., Über abstrakte Topologie, Monatschefte für Math. und Phys., 36, 219-258 (1929)
[7] Tietze, H., Über die Topologische Invariante mehr dimensionale Mannigfaltigkeiten, Monatschefte für Math. und Phys., 19, 1-118 (1907)
[8] Veblen, Oswald, Analysis Situs, (Cambridge Colloquium (1916), Amer. Math. Soc.) · Zbl 0001.40604
[9] Vietoris, L., Über den höheren Zusammenhang von Kompakten Räume und eine Klasse von Abbildungen, welche ihn ungeändert lässt, (Proc. Amsterdam, 29 (1926)), 1008-1013
[10] Vietoris, L., Über den höheren Zusammenhang Kompakter Räume, Jahresbericht der Deutsche Mathematiker-Ver., 36, 28-29 (1927), (Kursiv)
[11] Vietoris, L., Über den höheren Zusammenhang Kompakter Räume und eine Klasse von zusammenhangstreue Abbildungen, Math. Ann., 27, 454-472 (1927)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.