The set of paths in a space and its algebraic structure. A historical account
Annales de la faculté des sciences de Toulouse Mathématiques (2013)
- Volume: 22, Issue: 5, page 915-968
- ISSN: 0240-2963
Access Full Article
topAbstract
topHow to cite
topKrömer, Ralf. "The set of paths in a space and its algebraic structure. A historical account." Annales de la faculté des sciences de Toulouse Mathématiques 22.5 (2013): 915-968. <http://eudml.org/doc/275292>.
@article{Krömer2013,
abstract = {The present paper provides a test case for the significance of the historical category “structuralism” in the history of modern mathematics. We recapitulate the various approaches to the fundamental group present in Poincaré’s work and study how they were developed by the next generations in more “structuralist” manners. By contrasting this development with the late introduction and comparatively marginal use of the notion of fundamental groupoid and the even later consideration of equivalence relations finer than homotopy of paths (their implicit presence from the outset in the proof of the group property of the fundamental group notwithstanding), we encounter “delay” phenomena which are explained by focussing on the actual uses of a concept in mathematical discourse.},
author = {Krömer, Ralf},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
language = {eng},
month = {12},
number = {5},
pages = {915-968},
publisher = {Université Paul Sabatier, Toulouse},
title = {The set of paths in a space and its algebraic structure. A historical account},
url = {http://eudml.org/doc/275292},
volume = {22},
year = {2013},
}
TY - JOUR
AU - Krömer, Ralf
TI - The set of paths in a space and its algebraic structure. A historical account
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2013/12//
PB - Université Paul Sabatier, Toulouse
VL - 22
IS - 5
SP - 915
EP - 968
AB - The present paper provides a test case for the significance of the historical category “structuralism” in the history of modern mathematics. We recapitulate the various approaches to the fundamental group present in Poincaré’s work and study how they were developed by the next generations in more “structuralist” manners. By contrasting this development with the late introduction and comparatively marginal use of the notion of fundamental groupoid and the even later consideration of equivalence relations finer than homotopy of paths (their implicit presence from the outset in the proof of the group property of the fundamental group notwithstanding), we encounter “delay” phenomena which are explained by focussing on the actual uses of a concept in mathematical discourse.
LA - eng
UR - http://eudml.org/doc/275292
ER -
References
top- Alexander (J. W.).— A lemma on systems of knotted curves. Nat. Acad. Proc., 9, p. 93-95 (1923).
- Alexander (J. W.).— Topological invariants of manifolds. Nat. Acad. Proc., 10, p. 493-494 (1924).
- Alexander (J. W.).— Topological invariants of knots and links. Transactions A. M. S., 30, p. 275-306 (1928). Zbl54.0603.03MR1501429
- Alexander (J. W.) and Briggs (G. B.).— On types of knotted curves. Annals of Math., (2) 28, p. 562-586 (1927). Zbl53.0549.02MR1502807
- Artin (E.).— Theorie der Zöpfe. Abh. Math. Sem. Hamburg, 4, p. 47-72 (1925). Zbl51.0450.01MR3069440
- Barrett (J. W.).— Holonomy and path structures in general relativity and yang-mills theory. International Journal of Theoretical Physics, 30(9), p. 1171-1215, September (1991). Zbl0728.53055MR1122025
- Bourbaki (N.).— Éléments de mathématique. Part I. Les structures fondamentales de l’analyse. Livre II. Algèbre. Chapitre I. Structures algébriques. Hermann et Cie., Paris (1942). Zbl0098.02501MR11070
- Bourbaki (N.).— The architecture of mathematics. Am. Math. Monthly, 57(4), p. 221-232, (1950). Zbl0037.00209MR33787
- Brandt (H.).— Über eine Verallgemeinerung des Gruppenbegriffs. Math. Ann., 96, p. 360-366, (1926).
- Brechenmacher (F.).— Self-portraits with Évariste Galois (and the shadow of Camille Jordan). (Auto-portraits avec Évariste Galois (et l’ombre de Camille Jordan).). Rev. Hist. Math., 17(2), p. 271-371 (2011). Zbl1229.01120MR2883519
- Brouwer (L. E. J.).— Continuous one-one transformations of surfaces in themselves. Koninklijke Nederlandse Akademie van Wetenschapen Proceedings, 15, p. 352-360 (1912).
- Brown (R.).— Elements of modern topology. McGraw Hill (1968). Zbl0159.52201MR227979
- Brown (R.).— From groups to groupoids: A brief survey. Bull. Lond Math. Soc., 19, p. 113-134 (1987). Zbl0612.20032MR872125
- Cartan (E.).— Sur la structure des groupes de transformations finis et continus (Thèse). Nony, Paris (1894). Zbl0007.10204
- Cartan (E.).— Sur la déformation projective des surfaces. Ann. de l’Éc. Norm., 37, p. 259-356 (1920). Zbl47.0656.05MR1509228
- Cartan (E.).— Sur le problème général de la déformation. In Comptes rendus du congrès internat. des math., p. 397-406 (1920). Zbl48.0817.02
- Cartan (E.).— Les récentes généralisations de la notion d’espace. Darboux Bull, (1924). Zbl50.0589.01
- Cartan (E.).— Les groupes d’holonomie des espaces généralisés. Acta Math., 48, p. 1-42, (1926). Zbl52.0723.01
- Cartan (E.).— La théorie des groupes finis et continus et l’Analysis situs, volume 42 of Mémorial des sciences mathématiques. 1930. OEuvres 1, p. 1165-1225.
- Chandler (B.), Magnus (W.).— The history of combinatorial group theory: a case study in the history of ideas, volume 9 of Studies in the History of Mathematics and Physical Sciences. Springer, New York (1982). Zbl0498.20001MR680777
- Chevalley (C.).— L’arithmétique dans les algèbres de matrices. 33 p. (Exposés mathématiques XIV.). Actual. sci. industr. 323 (1936). Zbl0014.29006
- Chorlay (R.).— L’émergence du couple local/global dans les théories géométriques, de Bernhard Riemann à la théorie des faisceaux 1851-1953. PhD thesis, Paris VII, direction Christian Houzel (2007).
- Chorlay (R.).— From problems to structures: the Cousin problems and the emergence of the sheaf concept. Arch. Hist. Exact Sci., 64(1), p. 1-73 (2010). Zbl1198.01001MR2570308
- Corry (L.).— Modern algebra and the rise of mathematical structures, volume 17 of Science Network Historical Studies. Birkhäuser, Basel (1996). Zbl0858.01022MR1391720
- van Dantzig (D.).— Le groupe fondamental des groupes compacts abstraits. C. R., 196, p. 1156-1159 (1933). Zbl59.0145.01
- Dehn (M.).— Über die Topologie des dreidimensionalen Raumes. Mathematische Annalen, 69, p. 137-168 (1910). Zbl41.0543.01MR1511580
- Dehn (M.).— Die beiden Kleeblattschlingen. Math. Ann., 75, p. 1-12 (1914). MR1511799
- Dehn (M.), Heegaard (P.).— Analysis situs. In Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, volume 3 I i, p. 153-220. Teubner (1907).
- Dieudonné (J.).— A history of algebraic and differential topology 1900-1960. Birkhaeuser Verlag, Boston, MA etc. (1989). Zbl0673.55002MR995842
- Eilenberg (S.), Steenrod (N. E).— Foundations of algebraic topology. Princeton University Press (1952). Zbl0047.41402MR50886
- Epple (M.).— Die Entstehung der Knotentheorie. Kontexte und Konstruktionen einer modernen mathematischen Theorie. Vieweg, Braunschweig (1999). Zbl0972.57001MR1716305
- Epple (M.).— From quaternions to cosmology: Spaces of constant curvature, ca. 1873-1925. In ICM, volume III, p. 935-945 (2002). Zbl0997.01005MR1957592
- Fano (G.).— Kontinuierliche geometrische Gruppen. Die Gruppentheorie als geometrisches Einteilungsprinzip. Enzyklop. d. math. Wissensch., III 1, p. 289-388 (1907). Zbl38.0499.01
- Gray (J.).— Linear Differential Equations and Group Theory from Riemann to Poincaré. Birkhäuser, Boston (1986). Zbl0949.01001MR891402
- Gray (J.).— On the history of the riemann mapping theorem. Rend. Circ. Mat. Palermo, Suppl. 34, p. 47-94 (1994). Zbl0810.01005MR1295591
- Hawkins (T.).— Weyl and the topology of continuous groups. In [45], p. 169-198 (1999). Zbl0949.22001MR1674913
- Hawkins (T.).— Emergence of the theory of Lie groups. An Essay in the History of Mathematics, 1869-1926. Springer, New York (2000). Zbl0965.01001MR1771134
- Hilb (E.).— Lineare Differentialgleichungen im komplexen Gebiet. In Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, volume II B, chapter 5, p. 471-562. Leipzig (1915). Zbl45.0465.06
- Hoehnke (H.-J.).— 66 Jahre Brandtsches Gruppoid. In Heinrich Brandt 1886-1986, volume 47 of Wissenschaftliche Beiträge 1986, pages 15-79. Martin-Luther Universität Halle-Wittenberg, Halle/Saale (1986). MR857294
- Hopf (H.).— Vektorfelder in n-dimensionalen Mannigfaltigkeiten. Math. Ann. 96, 96, p. 225-250 (1926). Zbl52.0571.01
- Houzel (C.).— Les debuts de la théorie des faisceaux. In Kashiwara, Masaki and Schapira, Pierre: Sheaves on manifolds, volume 292 of Grundlehren der Mathematischen Wissenschaften, p. 7-22. Springer-Verlag, Berlin (1990). Zbl0709.18001MR1074006
- Houzel (C.).— Histoire de la théorie des faisceaux. In Jean-Michel Kantor, editor, Jean Leray (1906-1998), Gazette des Mathématiciens, supplément au numéro 84, pages 35-52. SMF (2000). Zbl1044.01532MR1775588
- Hurewicz (W.).— Beiträge zur Topologie der Deformationen I. Höherdimensionale Homotopiegruppen. Proceedings of the Koninklijke Akademie von Wetenschappen te Amsterdam. Section of Sciences, 38, p. 112-119, 1935. Seifert: ZBL.010.37801. Zbl0010.37801
- Hurewicz (W.).— Beiträge zur Topologie der Deformationen II. Homotopie- und Homologiegruppen. Proceedings of the Koninklijke Akademie von Wetenschappen te Amsterdam. Section of Sciences, 38, p. 521-528 (1935). Seifert: ZBL.011.37101. Zbl0011.37101
- James (I. M.) (ed.).— History of Topology. North-Holland, Amsterdam (1999). Zbl0922.54003MR1674906
- Jordan (C.).— Des contours tracés sur les surfaces. Journal de Mathématiques Pures et Appliquées, 9, p. 110-130 (1866). OEuvres 4, p. 91-112.
- Jordan (C.).— Sur la déformation des surfaces. Journal de Mathématiques Pures et Appliquées, 9, p. 105-109 (1866). OEuvres 4, p. 85-89.
- Jordan (C.).— Traité des substitutions et des équations algébriques. Paris (1870). Zbl0828.01011
- Jordan (C.).— Mémoire sur une application de la théorie des substitutions à l’étude des équations différentielles linéaires. Bulletin Soc. Math. France, 2, p. 100-127 (1873-1874). MR1503686
- Jordan (C.).— Sur une application de la théorie des substitutions aux équations différentielles linéaires. Comptes rendus Acad. Sciences Paris, 78, p. 741-743 (1874).
- Klein (F.).— Vorlesungen über die hypergeometrische Funktion (G¬ottingen). Teubner, Leipzig (1894).
- Kneser (H.).— Geschlossene Flächen in dreidimensionalen Mannigfaltigkeiten. Jahresbericht D. M. V., 38, p. 248-260 (1929). Zbl55.0311.03
- Krömer (R.).— La « machine de Grothendieck », se fonde-t-elle seulement sur des vocables métamathématiques? Bourbaki et les catégories au cours des années cinquante. Revue d’Histoire des Mathématiques, 12, p. 111-154 (2006). Zbl1177.01034
- Krömer (R.).— Tool and object. A history and philosophy of category theory, volume 32 of Science Network Historical Studies. Birkhäuser, Basel (2007). Zbl1114.18001MR2272843
- Krömer (R.).— Ein Mathematikerleben im 20. Jahrhundert. Zum 10. Todestag von Samuel Eilenberg. Mitteilungen der deutschen Mathematiker-Vereinigung, 16, p. 160-167 (2008). MR2522904
- Krömer (R.).— Are we still Babylonians? The structure of the foundations of mathematics from a wimsattian perspective. In L. Soler E. Trizio Th. Nickles W. Wimsatt, editor, Characterizing the Robustness of the Sciences After the Practical Turn of Philosophy of Science, volume 292 of Boston Studies in the Philosophy of Science. Springer (2012).
- Lefschetz (S.).— Topology, volume 12 of AMS Colloquium Publ. AMS, Providence/RI (1930).
- Mackaay (M.), Picken (R.).— The holonomy of gerbes with connections. Adv. Math., 170, p. 287-339 (2002). Zbl1034.53051MR1932333
- Marquis (J.-P.).— Some threads between homotopy theory and category theory: axiomatizing homotopy theories. Oberwolfach Reports, 08, p. 480-482 (2009).
- Milnor (J.).— Construction of universal bundles i. Annals Math., 63, p. 272-284 (1956). Zbl0071.17302MR77122
- Morse (H. M.).— Recurrent geodesics on a surface of negative curvature. American M. S. Trans., 22, p. 84-100 (1921). Zbl48.0786.06MR1501161
- Morse (H. M.).— A fundamental class of geodesics on any closed surface of genus greater than one. American M. S. Trans., 26, p. 25-60 (1924). Zbl50.0466.04MR1501263
- Newman (M.H.A.).— Elements of the topology of plane sets of points. University press, Cambridge, second edition (1951). Zbl0045.44003MR44820
- Nielsen (J.).— Die Isomorphismen der allgemeinen, unendlichen Gruppe mit zwei Erzeugenden. Math. Ann., 78, p. 385-397 (1917). Zbl46.0175.01MR1511907
- Nielsen (J.).— Über die Minimalzahl der Fixpunkte bei den Abbildungstypen der Ring ächen. Math. Ann., 82, p. 83-93 (1920). Zbl47.0527.03MR1511973
- Nielsen (J.).— Über fixpunktfreie topologische Abbildungen geschlossener Flächen. Math. Ann., 81, p. 94-96 (1920). Zbl47.0527.02MR1511960
- Nielsen (J.).— Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen. Acta Math., 50, p. 189-358 (1927). Zbl53.0545.12MR1555256
- Noether (M.), Wirtinger (W.) (ed.).— Bernhard Riemann’s Gesammelte mathematische Werke. Nachträge (1902).
- Patterson (S.).— Uniformisierung und diskontinuierliche Gruppen. In [109], p. 231-240. Teubner (1997). Zbl0873.01017
- Poincaré (H.).— Théorie des groupes fuchsiens. Acta Mathematica, 1, p. 1-62 (1882).
- Poincaré (H.).— Sur les groupes des équations linéaires. Comptes rendus hebdomadaires de l’Académie des sciences, 96, p. 691-694 (1883).
- Poincaré (H.).— Sur un théorème de la théorie générale des fonctions. Bulletin de la société mathématique de France, 11, p. 112-125 (1883).
- Poincaré (H.).— Sur les groupes des équations linéaires. Acta Mathematica, 4, p. 201-311, (1884).
- Poincaré (H.).— Analysis situs. Journal de l’École polytechnique, 1, p. 1-121 (1895). Zbl26.0541.07
- Poincaré (H.).— Cinquième complèment à l’analysis situs. Rendiconti del Circolo Matematico di Palermo, 18, p. 45-110 (1904). Zbl35.0504.13
- Poincaré (H.).— Sur l’uniformisation des fonctions analytiques. Acta Mathematica, 31, p. 1-63, (1908). Zbl38.0452.02MR1555036
- Poincaré (H.).— Œuvres, vol. 4. Gauthier-Villars, Paris (1950). Zbl0072.24103
- Poincaré (H.).— Œuvres, vol. 6. Gauthier-Villars, Paris (1953). Zbl0072.24103
- Poincaré (H.).— La correspondance entre Henri Poincaré et Gösta Mittag-Leffler (1999).
- Poincaré (H.).— Papers on Topology: Analysis Situs and Its Five Supplements. Translated by John Stillwell. AMS/LMS (2010). Zbl1204.55002MR2723194
- Pontrjagin (L.).— Sur les groupes topologiques compacts et le cinquième problème de M. Hilbert. Comptes rendus Acad. Sciences Paris, 198, p. 238-240 (1934). Zbl0008.24603
- Pontrjagin (L.).— Topological groups. Princeton University press, 1939. Translated from the russian by Emma Lehmer. Zbl65.0872.02
- Pontrjagin (L.).— Topological groups. Second edition. Gordon and Breach, New York (1966). Zbl62.0443.02MR201557
- Reidemeister (K.).— Knoten und Gruppen. Abh. Math. Sem. Hamburg, 5, p. 7-23 (1927). MR3069461
- Reidemeister (K.).— Über Knotengruppen. Abhandlungen Hamburg, 6, p. 56-64 (1928). MR3069488
- Reidemeister (K.).— Knoten und Verkettungen. Math. Z., 29, p. 713-729 (1929). MR1545033
- Reidemeister (K.).— Einführung in die kombinatorische Topologie. Vieweg, Braunschweig (1932). Zbl0042.17702
- Reidemeister (K.).— Überdeckungen von Komplexen. J. Reine Angew. Math., 173, p. 164-173, (1935). Zbl0012.12604
- Reidemeister (K.).— Fundamentalgruppen von Komplexen. Mathematische Zeitschrift, 40, p. 406-416 (1936). Zbl0012.22803MR1545568
- Sarkaria (K.S.).— The topological work of Henri Poincaré. In [45], p. 123-167 (1999). Zbl0959.54002MR1674912
- Schappacher (N.), Goldstein (C.), and Schwermer (J.)(ed.).— The shaping of arithmetic after C.F. Gauss’s Disquisitiones Arithmeticae. Springer (2007). Zbl1149.01001MR2308276
- Scholz (E.).— Geschichte des Mannigfaltigkeitsbegriffs von Riemann bis Poincaré. Birkhäuser, Boston, Basel, Stuttgart (1980). Zbl0438.01004MR631524
- Schreier (O.).— Abstrakte kontinuierliche Gruppen. Abh. Math. Sem. Hamburg, 4, p. 15-32, (1925). Zbl51.0112.04MR3069438
- Schreier (O.).— Die Verwandtschaft stetiger Gruppen im großen. Abh. Math. Sem. Hamburg, 5, p. 233-244 (1927). Zbl53.0110.02MR3069479
- Seifert (H.) and Threlfall (W.).— Lehrbuch der Topologie. Teubner, Leipzig (1934). Zbl0009.08601
- Spanier (E. H.).— Algebraic Topology, volume 11 of McGraw Hill series in higher mathematics. McGraw Hill (1966). Zbl0145.43303MR210112
- Steenrod (N. E.).— Topological methos for the construction of tensor functions. Annals Math., 43, p. 116-131 (1942). Zbl0061.41001MR5357
- Steenrod (N. E.).— Homology with local coefficients. Annals Math. (2), 44, p. 610-627 (1943). MR5,104f. Zbl0061.40901MR9114
- Steenrod (N. E.).— The topology of fibre bundles., volume 14 of Princeton Mathematical Series. Princeton University Press, Princeton (1951). Zbl0054.07103MR39258
- Threlfall (W.), Seifert (H.).— Topologische Untersuchung der Diskontinuitätsbereiche endlicher Bewegungsgruppen des dreidimensionalen sphärischen Raumes. Mathematische Annalen, 104, p. 1-70 (1930). Zbl56.1132.02
- Tietze (H.).— Über die topologischen Invarianten mehrdimensionaler Mannigfaltigkeiten. Monatsh. Math., 19, p. 1-118 (1908). Zbl39.0720.05MR1547755
- Vanden Eynde (R.).— Historical evolution of the concept of homotopic paths. Arch. Hist. Ex. Sci., pages 127-188 (1992). Zbl0766.01015MR1201632
- Vanden Eynde (R.).— Development of the concept of homotopy. In [45], pages 65-102 (1999). Zbl0944.55001MR1674909
- Veblen (O.).— The Cambridge Colloquium, 1916. Part II: Analysis Situs. (Amer. Math. Soc., Colloquium Lectures, vol. V.). New York: Amer. Math. Soc. VII u. 150 S. 8 0 (1922).
- Veblen (O.), Whitehead (J. H. W.).— The foundations of Differential Geometry, volume 29 of Cambridge Tracts in Mathematics and Mathematical Physics. Cambridge Univ. Press, London (1932).
- Volkert (K. T.).— Das Homöomorphieproblem, insbesondere der 3-Mannigfaltigkeiten, in der Topologie 1892-1935. Philosophia Scientiae, Cahier spécial 4 (2002).
- Weyl (H.).— Die Idee der Riemannschen Fläche.— Teubner, Leipzig (1913). Zbl0283.30023
- Weyl (H.).— On the foundations of infinitesimal geometry. Bulletin A. M. S., 35, p. 716-725 (1929). Zbl55.1027.01MR1561797
- Weyl (H.).— Die Idee der Riemannschen Fläche. Herausgegeben von Reinhold Remmert. Teubner, Stuttgart (1997). Zbl0283.30023MR1440406
- Weyl (H.).— The concept of a Riemann surface. Translated from the German by Gerald R. MacLane. Dover, 3rd edition (2009). MR166351
- Whitney (H.).— Differentiable manifolds. Ann. Math., 37, p. 645-680 (1936). Zbl0015.32001MR1503303
- Wussing (H.).— Die Genesis des abstrakten Gruppenbegriffs. Berlin (1969). Zbl0199.29101
NotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.