Inventiones mathematicae

, Volume 219, Issue 1, pp 39–73 | Cite as

Motivic spheres and the image of the Suslin–Hurewicz map

  • Aravind AsokEmail author
  • Jean Fasel
  • Ben Williams


We show that an old conjecture of A. A. Suslin characterizing the image of a Hurewicz map from Quillen K-theory in degree n to Milnor K-theory in degree n admits an interpretation in terms of unstable \({\mathbb {A}}^1\)-homotopy sheaves of the general linear group. Using this identification, we establish Suslin’s conjecture in degree 5 for infinite fields having characteristic unequal to 2 or 3. We do this by linking the relevant unstable \({\mathbb {A}}^1\)-homotopy sheaf of the general linear group to the stable \({\mathbb {A}}^1\)-homotopy of motivic spheres.



The first author would like to thank Sasha Merkurjev for explaining Suslin’s approach to his eponymous conjecture in degree 4; even though this makes no appearance here, it was still an essential input. The authors would also like to thank Marc Levine and the University of Duisburg-Essen where the initial idea of this paper was conceived and Kirsten Wickelgren for her collaboration in an early stage of this project. Finally, the authors would like to thank Oliver Röndigs for helpful discussions about [31], and for comments and corrections on a draft of this work.


  1. 1.
    Asok, A., Fasel, J.: Algebraic vector bundles on spheres. J. Topol. 7(3), 894–926 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Asok, A., Fasel, J.: A cohomological classification of vector bundles on smooth affine threefolds. Duke Math. J. 163(14), 2561–2601 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Asok, A., Fasel, J.: Splitting vector bundles outside the stable range and \(\mathbb{A}^1\)-homotopy sheaves of punctured affine spaces. J. Am. Math. Soc. 28(4), 1031–1062 (2015)zbMATHCrossRefGoogle Scholar
  4. 4.
    Asok, A., Fasel, J.: An explicit \(KO\)-degree map and applications. J. Topol. 10(1), 268–300 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Asok, A., Hoyois, M., Wendt, M.: Affine representability results in \(\mathbb{A}^1\)-homotopy theory, I: vector bundles. Duke Math. J. 166(10), 1923–1953 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Asok, A., Hoyois, M., Wendt, M.: Affine representability results in \(\mathbb{A}^1\)-homotopy theory, II: principal bundles and homogeneous spaces. Geom. Topol. 22(2), 1181–1225 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Arason, J.K.: Cohomologische invarianten quadratischer Formen. J. Algebra 36(3), 448–491 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Asok, A., Wickelgren, K., Williams, T.B.: The simplicial suspension sequence in \(\mathbb{A}^1\)-homotopy. Geom. Topol. 21(4), 2093–2160 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Bousfield, A.K., Kan, D.M.: Homotopy Limits, Completions and Localizations. Lecture Notes in Mathematics, vol. 304. Springer, Berlin (1972)zbMATHCrossRefGoogle Scholar
  10. 10.
    Dugger, D., Isaksen, D.C.: Topological hypercovers and \(\mathbb{A}^1\)-realizations. Math. Z. 246(4), 667–689 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Gersten, S.M.: Higher \(K\)-Theory of Rings. Lecture Notes in Mathematics, vol. 341, pp. 3–42. Springer, Berlin (1973)zbMATHCrossRefGoogle Scholar
  12. 12.
    Guin, D.: Homologie du groupe linéaire et \(K\)-théorie de Milnor des anneaux. J. Algebra 123(1), 27–59 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Hutchinson, K., Wendt, M.: On the third homology of \(SL_2\) and weak homotopy invariance. Trans. Am. Math. Soc. 367(10), 7481–7513 (2015)zbMATHCrossRefGoogle Scholar
  14. 14.
    Isaksen, D.C.: Etale realization on the \(\mathbb{A}^1\)-homotopy theory of schemes. Adv. Math. 184(1), 37–63 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Jacob, B., Rost, M.: Degree four cohomological invariants for quadratic forms. Invent. Math. 96(3), 551–570 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Kerz, M.: The Gersten conjecture for Milnor \(K\)-theory. Invent. Math. 175(1), 1–33 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Knebusch, M., Rosenberg, A., Ware, R.: Structure of witt rings and quotients of abelian group rings. Am. J. Math. 94(1), 119–155 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Lam, T.Y.: Introduction to Quadratic Forms Over Fields. Graduate Studies in Mathematics, vol. 67. American Mathematical Society, Providence (2005)zbMATHGoogle Scholar
  19. 19.
    Morel, F.: Théorie homotopique des schémas. Astérisque, (256):vi+119, (1999)Google Scholar
  20. 20.
    Morel, F.: An Introduction to \(\mathbb{A}^1\)-Homotopy Theory Contemporary Developments in Algebraic \(K\)-Theory. ICTP Lecture Notes, vol. 15, pp. 357–441. Abdus Salam International Centre for Theoretical Physics, Trieste (2004)Google Scholar
  21. 21.
    Morel, F.: Milnor’s conjecture on quadratic forms and mod 2 motivic complexes. Rend. Sem. Mat. Univ. Padova 114(63–101), 2005 (2006)MathSciNetGoogle Scholar
  22. 22.
    Morel, F.: On the Friedlander–Milnor conjecture for groups of small rank. In: Jerison, D., Mazur, B., Mrowka, T., Schmid, W., Stanley, R., Yau, S.T. (eds.) Current Developments in Mathematics, 2010, pp. 45–93. International Press, Somerville (2011)Google Scholar
  23. 23.
    Morel, F.: \({\mathbb{A}}^1\)-algebraic topology over a field. Lecture Notes in Mathematics, vol. 2052. Springer, Heidelberg (2012)zbMATHCrossRefGoogle Scholar
  24. 24.
    Merkurjev, A.S., Suslin, A.A.: The group \(K_3\) for a field. Izv. Akad. Nauk SSSR Ser. Mat. 54(3), 522–545 (1990)Google Scholar
  25. 25.
    Morel, F., Voevodsky, V.: \({\mathbf{A}}^1\)-homotopy theory of schemes. Inst. Hautes Études Sci. Publ. Math. 90(45–143), 1999 (2001)Google Scholar
  26. 26.
    Nesterenko, YuP, Suslin, A.A.: Homology of the general linear group over a local ring, and Milnor’s \(K\)-theory. Izv. Akad. Nauk SSSR Ser. Mat. 53(1), 121–146 (1989)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Orlov, D., Vishik, A., Voevodsky, V.: An exact sequence for \(K^M_\ast /2\) with applications to quadratic forms. Ann. Math. (2) 165(1), 1–13 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Rector, D.L.: \(K\)-theory of a space with coefficients in a (discrete) ring. Bull. Am. Math. Soc. 77, 571–575 (1971)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Röndigs, O., Østvær, P.-A.: Slices of hermitian \(K\)-theory and Milnor’s conjecture on quadratic forms. Geom. Topol. 20(2), 1157–1212 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Rost, M.: Hilbert \(90\) for \(K_3\) for Degree-Two Extensions. (Preprint), (1986)Google Scholar
  31. 31.
    Röndigs, O., Spitzweck, M., Østvær, P.-A.: The first stable homotopy groups of motivic spheres. Ann. Math. (2) 189(1), 1–74 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Schlichting, M.: Euler class groups and the homology of elementary and special linear groups. Adv. Math. 320, 1–81 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Schlichting, M., Tripathi, G.S.: Geometric models for higher Grothendieck-Witt groups in \(\mathbb{A}^1\)-homotopy theory. Math. Ann. 362(3–4), 1143–1167 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Suslin, A. A.: Homology of \({\rm GL}_{n}\), characteristic classes and Milnor \(K\)-theory. In Algebraic \(K\)-Theory, Number Theory, Geometry and Analysis (Bielefeld, 1982), volume 1046 of Lecture Notes in Mathematics, pp. 357–375. Springer, Berlin, (1984)Google Scholar
  35. 35.
    Weibel, C.A.: The \(K\)-Book, volume 145 of Graduate Studies in Mathematics. American Mathematical Society, Providence (2013)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Southern CaliforniaLos AngelesUSA
  2. 2.Institut Fourier - UMR 5582Université Grenoble AlpesGrenoble Cedex 9France
  3. 3.Department of MathematicsThe University of British ColumbiaVancouverCanada

Personalised recommendations