Skip to main content
SpringerLink
Log in

Milnor’s Work in Algebra and Its Ramifications

  • Chapter
  • First Online:
The Abel Prize 2008-2012

Part of the book series: The Abel Prize ((AP))

  • 2288 Accesses

Abstract

Highlights of Milnor’s extensive work in algebra are surveyed, with an emphasis on their ramifications and influence on further developments in the field. After brief discussion of his work on Hopf algebras, and on Growth of groups, more substantial treatment is given of his work on The Congruence Subgroup Problem, and on Algebraic K-theory and quadratic forms.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 139.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 179.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 179.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    When \(\operatorname{char}(\mathbf {F}) = 2\) there are natural analogues of \(h_{n}^{\mathbf {F}}\) and MHF(n), but with H n(G,μ 2) replaced by groups H n(F) defined in terms of differentials. (See [17, p. 5].)

References

  1. Arason, J., Pfister, A.: Beweis des Krullschen Durchschnittsatzes für den Wittring. Invent. Math. 12, 173–176 (1971)

    Article  MathSciNet  Google Scholar 

  2. Bass, H., Lazard, M., Serre, J.-P.: Sous-groupes d’indices finis dans \(\operatorname{SL}(n,Z)\). Bull. Am. Math. Soc. 70, 385–392 (1964)

    Article  MathSciNet  Google Scholar 

  3. Bass, H., Milnor, J., Serre, J.-P.: Solution of the congruence subgroup problem for \(\operatorname{SL}_{n}\) (n≥3) and \(\operatorname{Sp}_{2n}\) (n≥2). Publ. Math. IHES 33, 59–137 (1967) (Erratum: On a functorial property of power residue symbols. Publ. Math. IHES 44 (1974), 241–244)

    Article  MathSciNet  Google Scholar 

  4. Bass, H.: Algebraic K-Theory. Benjamin, New York (1968)

    MATH  Google Scholar 

  5. Bass, H.: On the degree of growth of a finitely generated nilpotent group. Proc. Lond. Math. Soc. 25, 603–614 (1972)

    Article  MathSciNet  Google Scholar 

  6. Bass, H., Tate, J.: The Milnor ring of a global field. Springer LNM, vol. 342, pp. 349–446 (1973)

    MATH  Google Scholar 

  7. Bass, H.: John Milnor the algebraist. In: Topological Methods in Modern Mathematics. A Symposium in Honor of John Milnor’s Sixtieth Birthday, pp. 45–83 (1993). Publish or Perish

    Google Scholar 

  8. Chevalley, C.: Deux théoèmes d’arithmétique. J. Math. Soc. Jpn. 3, 36–44 (1951)

    Article  Google Scholar 

  9. Grigorchuk, R.I.: On the Milnor problem of group growth. Dokl. Akad. Nauk SSSR 271, 30–33 (1983)

    MathSciNet  Google Scholar 

  10. Gromov, M.: Groups of polynomial growth and expanding maps. Publ. Math. IHES 53, 53–73 (1981)

    Article  MathSciNet  Google Scholar 

  11. Kato, K.: Symmetric bilinear forms, quadratic forms and Milnor K-theory in characteristic two. Invent. Math. 66(3), 493–510 (1982)

    Article  MathSciNet  Google Scholar 

  12. Kneser, M.: Strong approximation, I, II. Algebraic groups and discontinuous subgroups. In: Proc. Symp. Pure Math. Amer Math. Soc., vol. IX, pp. 187–196 (1966)

    Google Scholar 

  13. Lubotzky, A.: Subgroup growth and congruence subgroups. Invent. Math. 119, 267–295 (1995)

    Article  MathSciNet  Google Scholar 

  14. Matsumoto, H.: Sur les sous-groupes arithmétiques des groupes semi-simples déployés. Ann. Sci. Éc. Norm. Super. 2, 1–62 (1969)

    Article  Google Scholar 

  15. Mennicke, J.: Finite factor groups of the unimodular group. Ann. Math. 81 (1965)

    Article  MathSciNet  Google Scholar 

  16. Merkurjev, A.S.: On the norm residue symbol of degree 2. Dokl. Akad. Nauk SSSR 261(3), 542–547 (1981)

    MathSciNet  Google Scholar 

  17. Merkurjev, A.S.: Developments in algebraic K-theory and quadratic forms after the work of Milnor. In: Collected Papers of John Milnor, V. Algebra, pp. 399–417. Am. Math. Soc., Providence (2010)

    Google Scholar 

  18. Milnor, J.: The Steenrod algebra and its dual. Ann. of Math. (2) 67, 150–171 (1958)

    Article  MathSciNet  Google Scholar 

  19. Milnor, J., Moore, J.C.: On the structure of Hopf algebras. Ann. Math. 81, 211–264 (1965)

    Article  MathSciNet  Google Scholar 

  20. Milnor, J.: Whitehead torsion. Bull. Amer. Math. Soc. (N. S.) 72, 358–426 (1966)

    Article  MathSciNet  Google Scholar 

  21. Milnor, J.: A note on curvature and the fundamental group. J. Differ. Geom. 2, 1–7 (1968)

    Article  MathSciNet  Google Scholar 

  22. Milnor, J.: Growth of finitely generated solvable groups. J. Differ. Geom. 2, 447–449 (1968)

    Article  MathSciNet  Google Scholar 

  23. Milnor, J.: Advanced problem 5603. MAA Monthly 75, 685–686 (1968)

    Google Scholar 

  24. Milnor, J.: Algebraic K-theory and quadratic forms. Invent. Math. 9, 318–344 (1969/1970)

    Article  MathSciNet  Google Scholar 

  25. Milnor, J.: Symmetric inner products in characteristic 2. Prospects in mathematics. In: Proc. Sympos., Princeton Univ., Princeton, NJ, 1970. Ann. of Math. Studies, vol. 70, pp. 59–75. Princeton Univ. Press, Princeton (1971)

    Google Scholar 

  26. Milnor, J.: Introduction to Algebraic K-Theory. Annals of Mathematics Studies, vol. 72. Princeton University Press, Princeton (1971)

    MATH  Google Scholar 

  27. Milnor, J.: Collected papers of John Milnor, v. In: Bass, H., Lam, T.-Y. (eds.) Algebra. Am. Math. Soc., Providence (2010)

    MATH  Google Scholar 

  28. Moore, C.: Group extensions of p-adic and adèlic groups. Publ. Math. IHES 35, 5–70 (1968)

    Article  Google Scholar 

  29. Orlov, D., Vishik, A., Voevodsky, V.: An exact sequence for \(K_{\ast}^{M}/2\) with applications to quadratic forms. Ann. of Math. (2) 165(1), 1–13 (2007)

    Article  MathSciNet  Google Scholar 

  30. Platonov, V.P., Rapinchuk, A.S.: Abstract properties of S-arithmetic groups and the congruence subgroup problem. Russian Acad. Sci. Izv. Math. 40, 455–476 (1993)

    MathSciNet  MATH  Google Scholar 

  31. Prasad, G., Rapinchuk, A.S.: Developments of the congruence subgroup problem after the work of Bass, Milnor, and Serre. In: Bass, H., Lam, T.-Y. (eds.) Collected Papers of John Milnor, V. Algebra. Am. Math. Soc., Providence (2010)

    Google Scholar 

  32. Quillen, D.: In: Higher Algebraic K-Theory I. Lecture Notes in Math., vol. 341, pp. 85–147. Springer, Berlin (1973)

    Chapter  Google Scholar 

  33. Raghunathan, M.S.: On the congruence subgroup problem. Publ. Math. IHES 46, 107–161 (1976)

    Article  MathSciNet  Google Scholar 

  34. Raghunathan, M.S.: On the congruence subgroup problem II. Invent. Math. 85, 73–117 (1986)

    Article  MathSciNet  Google Scholar 

  35. Rost, M.: On Hilbert Satz 90 for K 3 for degree-two extensions, available as http://www.mathematik.uni-bielefeld.de/~rost/K3-86.html (1986)

  36. Serre, J.-P.: Sur les groupes de congruence des variétés abéliennes. Izv. Akad. Nauk SSSR, Ser. Mat. 28, 3–18 (1964)

    MathSciNet  MATH  Google Scholar 

  37. Serre, J.-P.: Le probème des groupes de congruence pour \(\operatorname{SL}_{2}\). Ann. Math. 92, 489–527 (1970)

    Article  MathSciNet  Google Scholar 

  38. Serre, J.-P.: Sur les groupes de congruence des variétés abéliennes II. Izv. Akad. Nauk SSSR, Ser. Mat. 35, 731–735 (1971)

    MathSciNet  MATH  Google Scholar 

  39. Steinberg, R.: Générateurs, relations et revêtements de groupes algébriques. In: Colloq. Théorie des Groupes Algébriques, Bruxelles, Librairie Universitaire, Louvain, 1962, pp. 113–127. Gauthier-Villars, Paris (1962)

    Google Scholar 

  40. Suslin, A., Voevodsky, V.: Bloch–Kato conjecture and motivic cohomology with finite coefficients. In: The Arithmetic and Geometry of Algebraic Cycles, Banff, AB, 1998. NATO Sci. Ser. C Math. Phys. Sci., vol. 548, pp. 117–189. Kluwer Academic, Dordrecht (2000)

    Chapter  Google Scholar 

  41. Tits, J.: Free subgroups of linear groups. J. Algebra 20, 250–270 (1972)

    Article  MathSciNet  Google Scholar 

  42. Voevodsky, V.: Triangulated categories of motives over a field. In: Cycles, Transfers, and Motivic Homology Theories. Ann. of Math. Stud., vol. 143, pp. 188–238. Princeton Univ. Press, Princeton (2000)

    MATH  Google Scholar 

  43. Voevodsky, V.: Motivic cohomology with Z/2-coefficients. IHES Math. Publ. 98, 59–104 (2003)

    Article  MathSciNet  Google Scholar 

  44. Voevodsky, V.: Reduced power operations in motivic cohomology. Publ. Math. IHES 98, 1–57 (2003)

    Article  MathSciNet  Google Scholar 

  45. Wolf, J.A.: Growth of finitely generated solvable groups, and curvature of Riemannian manifolds. J. Differ. Geom. 2, 421–446 (1968)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Michigan Ann Arbor, 3864 East Hall, 525 E. University, Ann Arbor, MI, 48109-1109, USA

    Hyman Bass

Authors
  1. Hyman Bass
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hyman Bass .

Editor information

Editors and Affiliations

  1. Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway

    Helge Holden

  2. Department of Mathematics, University of Oslo, Oslo, Norway

    Ragni Piene

1 Electronic Supplementary Material

Below are the links to the electronic supplementary material.

A lecture by Prof. Ghys in connection with the Abel Prize 2011 to John Milnor (MP4 399 MB)

A lecture by Prof. Hopkins in connection with the Abel Prize 2011 to John Milnor (MP4 306 MB)

A lecture by Prof. McMullen in connection with the Abel Prize 2011 to John Milnor (MP4 362 MB)

The Abel Lecture by John Milnor, the Abel Laureate 2011 (MP4 379 MB)

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Bass, H. (2014). Milnor’s Work in Algebra and Its Ramifications. In: Holden, H., Piene, R. (eds) The Abel Prize 2008-2012. The Abel Prize. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39449-2_19

Download citation

Publish with us

Policies and ethics

Search

Navigation