The first stable homotopy groups of motivic spheres

Abstract

We compute the $1$-line of stable homotopy groups of motivic spheres over fields of characteristic not two in terms of hermitian and Milnor $K$-groups. This is achieved by solving questions about convergence and differentials in the slice spectral sequence.

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      author = {Hu, P. and Kriz, I. and Ormsby, K.},
      title = {Convergence of the motivic {A}dams spectral sequence},
      journal = {J. $K$-{T}heory},
      fjournal = {Journal of $K$-Theory. $K$-Theory and its {A}pplications in {A}lgebra, {G}eometry, {A}nalysis \& {T}opology},
      volume = {7},
      year = {2011},
      number = {3},
      pages = {573--596},
      issn = {1865-2433},
      mrclass = {14F42 (55T15)},
      mrnumber = {2811716},
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      doi = {10.1017/is011003012jkt150},
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      title = {The indecomposable {$K_3$} of fields},
      journal = {Ann. Sci. École Norm. Sup. (4)},
      fjournal = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
      volume = {22},
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      issn = {0012-9593},
      mrclass = {11R70 (12G05 18F25 19D55 19F27)},
      mrnumber = {1005161},
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      url = {http://www.numdam.org/item?id=ASENS_1989_4_22_2_255_0},
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      mrclass = {14C25 (14F42 19E08 19E15 55P42)},
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      doi = {10.1112/jtopol/jtm004},
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      mrclass = {14C25 (14F42 19E08 19E15 55P42)},
      mrnumber = {3217623},
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      author = {Röndigs, Oliver},
      title = {On the $\eta$-inverted sphere},
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    @ARTICLE{RO1,
      author = {Röndigs, Oliver and {\O}stv{æ}r, Paul Arne},
      title = {Motives and modules over motivic cohomology},
      journal = {C. R. Math. Acad. Sci. Paris},
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      volume = {342},
      year = {2006},
      number = {10},
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      issn = {1631-073X},
      mrclass = {14F42},
      mrnumber = {2227753},
      doi = {10.1016/j.crma.2006.03.013},
      url = {https://doi.org/10.1016/j.crma.2006.03.013},
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    @ARTICLE{RO2,
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      title = {Modules over motivic cohomology},
      journal = {Adv. Math.},
      fjournal = {Advances in Mathematics},
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      mrclass = {14F42 (55P42)},
      mrnumber = {2399164},
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      doi = {10.1007/s00208-008-0208-5},
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      author = {Röndigs, Oliver and {\O}stv{æ}r, Paul Arne},
      title = {The multiplicative structure on the graded slices of {H}ermitian {$K$}-theory and {W}itt-theory},
      journal = {Homology Homotopy Appl.},
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      mrclass = {19G38 (14F42 19G12)},
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      title = {Slices of {H}ermitian {$K$}-theory and {M}ilnor's conjecture on quadratic forms},
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      mrclass = {19G38 (11E04 14F42 55P42 55T05)},
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      note = {to appear in the \emph{{P}roceedings of the International Colloquium on $K$-Theory, TIFR} (2016)},
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      title = {A possible new approach to the motivic spectral sequence for algebraic {$K$}-theory},
      booktitle = {Recent Progress in Homotopy Theory},
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      volume = {246},
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      mrclass = {14F42 (55P42)},
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      }

Authors

Oliver Röndigs

Institut für Mathematik, Universität Osnabrück, Germany

Markus Spitzweck

Institut für Mathematik, Universität Osnabrück, Germany

Paul Arne Ostvær

Department of Mathematics, University of Oslo, Norway