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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
Article
  • 103 Downloads

Abstract

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.

Notes

Acknowledgements

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.

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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

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