A Brief History of Algebraic Topology

  • Mahima Ranjan Adhikari


This chapter focuses the history on the emergence of the ideas leading to new areas of study in algebraic topology and conveys the contributions of some mathematicians who introduced new concepts or proved theorems of fundamental importance or inaugurated new theories in algebraic topology starting from the creation of homotopy, fundamental group, and homology group by H. Poincaré (1854–1912) in 1895, which are the first most profound and far reaching inventions in algebraic topology.


Vector Bundle Fundamental Group Simplicial Complex Homology Group Homotopy Class 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. Adams, J.F.: On the non-existence of elements of Hopf invariant one. Ann. Math. 72, 20–104 (1960)MathSciNetCrossRefMATHGoogle Scholar
  2. Adams, J.F.: Vector fields on spheres. Ann. Math. 75, 603–632 (1962)MathSciNetCrossRefMATHGoogle Scholar
  3. Adams, J.F.: Stable Homotopy Theory. Springer, Berlin (1964)CrossRefMATHGoogle Scholar
  4. Adams, J.F.: Algebraic Topology: A Student’s Guide. Cambridge University Press, Cambridge (1972)CrossRefMATHGoogle Scholar
  5. Alexander, J.W.: On the chains of a complex and their duals. Proc. Natl. Acad. Sci., USA 21, 509–511 (1935)Google Scholar
  6. Atiyah, M.F.: K-theory. Benjamin, New York (1967)MATHGoogle Scholar
  7. Atiyah, M.F., Hirzebruch, F.: Riemann-Roch theorems of differentiable manifolds. Bull. Am. Math. Soc. 65, 276–281 (1959)Google Scholar
  8. Atiyah, M.F., Hirzebruch F.: Vector bundles and homogeneous spaces. Proc. Symp. Pure Math., Am. Math. Soc. 7–38 (1961)Google Scholar
  9. Aull, C.E., Lowen R. (eds.): Handbook of the History of General Topology, vol. 3. Kluwer Academic Publishers, Berlin (2001)Google Scholar
  10. Barratt, M.G.: Track groups I. Proc. Lond. Math. Soc. 5(3), 71–106 (1955)Google Scholar
  11. Borsuk, K.: Sur les groupes des classes de transformations continues, pp. 1400–1403. C.R. Acad. Sci, Paris (1936)MATHGoogle Scholar
  12. Bott, R.: The stable homotopy of the classical groups. Ann. Math. 70, 313–337 (1959)MathSciNetCrossRefMATHGoogle Scholar
  13. Brown, E.H.: Cohomology theories. Ann. Math. 75, 467–484 (1962)MathSciNetCrossRefMATHGoogle Scholar
  14. Čech, E.: Théorie génerale de l’homologie dans un espace quelconque’. Fund. Math. 19, 149–183 (1932)Google Scholar
  15. Dieudonné, J.: A History of Algebraic and Differential Topology, 1900–1960. Modern Birkhäuser, Boston (1989)Google Scholar
  16. Dold, A.: Relations between ordinary and extraordinary homology. Algebr. Topol. Colloq. Aarhus 2–9 (1962)Google Scholar
  17. Dold, A., Thom, R.: Quasifaserungen und unendliche symmetrische Produkte. Ann. Math. Second Series 67, 239–281 (1958)Google Scholar
  18. Eilenberg, S., MacLane, S.: Natural isomorphism in group theory. Proc. Natl. Acad. Sci. USA 28, 537–544 (1942)Google Scholar
  19. Eilenberg, S., MacLane, S.: General theory of natural equivalence. Trans. Am. Math. Soc. 58, 231–294 (1945a)Google Scholar
  20. Eilenberg, S., MacLane, S.: Relations between homology and homotopy groups of spaces. Ann. Math. 46(2) (1945b)Google Scholar
  21. Eilenberg, S., Steenrod, N.: Axiomatic approach to homology theory. Proc. Natl. Acad. Sci. USA 31, 117–120 (1945)MathSciNetCrossRefMATHGoogle Scholar
  22. Eilenberg, S., Steenrod, N.: Foundations of Algebraic Topology. Princeton University Press, Princeton (1952)Google Scholar
  23. Freedman, M.: The topology of four-dimensional manifolds. J. Differ. Geom. 17, 357–453 (1982)MathSciNetMATHGoogle Scholar
  24. Freudenthal, H.: Über die Klassen von Sphärenabbildungen. Compositio Mathematica 5, 299–314 (1937)MathSciNetMATHGoogle Scholar
  25. Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)Google Scholar
  26. Hilton, P.J.: An Introduction to Homotopy Theory. Cambridge University Press, Cambridge (1983)Google Scholar
  27. Hilton, P.J., Wylie, S.: Homology Theory. Cambridge University Press, Cambridge (1960)CrossRefMATHGoogle Scholar
  28. Hopf, H.: Über die Abbildungen der 3-Sphäre auf die Kugelfleche. Math. Ann. 104, 637–665 (1931)MathSciNetCrossRefMATHGoogle Scholar
  29. Hopf, H.: Über die Abbildungen von Sphären auf Sphären niedrigerer Dimension. Fund. Math. 25, 427–440 (1935)MATHGoogle Scholar
  30. Hurewicz, W.: Beitrage der Topologie der Deformationen. Proc. K. Akad. Wet., Ser. A 38, 112–119, 521–528 (1935)Google Scholar
  31. Hurewicz, W.: On duality theorems. Bull. Am. Math. Soc. 47, 562–563 (1941)Google Scholar
  32. Hurewicz, W.: On the concept of fibre space. Proc. Natl. Acad. Sci., USA 41, 956–961 (1955)Google Scholar
  33. James, I.M.: (ed.) History of Topology. North-Holland, Amsterdam (1999)Google Scholar
  34. Kampen van. E.: On the connection between the fundamental groups of some related spaces. Am. J. Math. 55, 261–267 (1933)Google Scholar
  35. Maunder, C.R.F.: Algebraic Topology. Van Nostrand Reinhold Company, London (1970)MATHGoogle Scholar
  36. Mayer, J.: Algebraic Topology. Prentice-Hall, New Jersey (1972)Google Scholar
  37. Milnor, J.W.: Spaces having the homotopy type of a \(CW\)-complex. Trans. Am. Math. Soc. 90, 272–280 (1959)MathSciNetMATHGoogle Scholar
  38. Olum, P.: Obstructions to extensions and homotopies. Ann. Math. 25, 1–50 (1950)MathSciNetCrossRefMATHGoogle Scholar
  39. Poincaré, H.: Analysis situs. J. Ecole polytech 1(2), 1–121 (1895)Google Scholar
  40. Poincaré, H.: Second complément \(\grave{a}\) l’analysis situs. Proc. Lond. Math. Soc. 32, 277–308 (1900)CrossRefMATHGoogle Scholar
  41. Poincaré, H.: Cinquiéme, complément \(\grave{a}\) l’analysis situs. Rc. Cir. Mat. Palermo. 18, 45–110 (1904)CrossRefMATHGoogle Scholar
  42. Poincaré, H.: Papers on topology: analysis situs and its five supplements (Translated by J. Stillwell, History of Mathematics). Am. Math. Soc. 37 (2010)Google Scholar
  43. Pontryagin, L.: The general topological theorem of duality for closed sets. Ann. Math. 35(2), 904–914 (1934)MathSciNetCrossRefMATHGoogle Scholar
  44. Smale, S.: Generalized Poincaré’s conjecture in dimensions greater than 4. Ann. Math. 74(2), 391–406 (1961)MathSciNetCrossRefMATHGoogle Scholar
  45. Spanier, E.H.: Borsuk’s cohomotopy groups. Ann. Math. 50, 203–245 (1949)MathSciNetCrossRefMATHGoogle Scholar
  46. Spanier, E.: Infinite symmetric products, function spaces and duality. Ann. Math. 69, 142–198 (1959)MathSciNetCrossRefMATHGoogle Scholar
  47. Spanier, E.: Algebraic Topology. McGraw-Hill, New York (1966)Google Scholar
  48. Stallings, J.R.: On fibering certain 3-manifolds, topology of 3-manifolds and related topics. In: Proceedings of The University of Georgia Institute, 1961, pp. 95–100. Prentice Hall, Englewood Cliffs (1962)Google Scholar
  49. Steenrod, N.E.: The Topology of Fibre Bundles. Princeton University Press, Princeton (1951)Google Scholar
  50. Steenrod, N.E.: Chohomology operations and obstructions to extending continuous functions. Adv. Math. 8, 371–416 (1972)MathSciNetCrossRefMATHGoogle Scholar
  51. Switzer, R.M.: Algebraic Topology-Homotopy and Homology. Springer, Berlin (1975)CrossRefMATHGoogle Scholar
  52. Whitehead, J.H.C.: On adding relations to homotopy groups. Ann. Math. 42, 409–428 (1941)MathSciNetCrossRefMATHGoogle Scholar
  53. Whitehead, J.H.C.: Combinatorial homotopy: I. Bull. Am. Math. Soc. 55, 213–245 (1949)MathSciNetCrossRefMATHGoogle Scholar
  54. Whitehead, J.H.C.: A certain exact sequence. Ann. Math. 52(52), 51–110 (1950)MathSciNetCrossRefMATHGoogle Scholar
  55. Whitehead, G.W.: A generalization of the Hopf invariant. Ann. Math. 51, 192–237 (1950)MathSciNetCrossRefMATHGoogle Scholar
  56. Whitehead, G.W.: On the Freudenthal theorem. Ann. Math. 57, 209–228 (1953)MathSciNetCrossRefMATHGoogle Scholar
  57. Whitehead, G.W.: Generalized homology theories. Trans. Am. Math. Soc. 102, 227–283 (1962)MathSciNetCrossRefMATHGoogle Scholar
  58. Zeeman, C.: The generalised Poincaré conjecture. Bull. Am. Math. Soc. 67, 270 (1961)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer India 2016

Authors and Affiliations

  1. 1.Institute for Mathematics, Bioinformatics, Information Technology and Computer Science (IMBIC)KolkataIndia

Personalised recommendations