Geodesic
The η–inverted ℝ–motivic sphere
Guillou, Bertrand1 ; Isaksen, Daniel2
1Department of Mathematics, University of Kentucky, 715 Patterson Office Tower, Lexington, KY 40506-0027, United States
2Department of Mathematics, Wayne State University, 656 W Kirby, Detroit, MI 48202, United States
Algebraic and Geometric Topology, Tome 16 (2016) no. 5, pp. 3005-3027
Cet article a éte moissonné depuis la source Mathematical Sciences Publishers

Voir la notice de l'article

We use an Adams spectral sequence to calculate the ℝ–motivic stable homotopy groups after inverting η. The first step is to apply a Bockstein spectral sequence in order to obtain h1 –inverted ℝ–motivic Ext groups, which serve as the input to the η–inverted ℝ–motivic Adams spectral sequence. The second step is to analyze Adams differentials. The final answer is that the Milnor–Witt (4k−1)–stem has order 2u+1, where u is the 2–adic valuation of 4k. This answer is reminiscent of the classical image of J. We also explore some of the Toda bracket structure of the η–inverted ℝ–motivic stable homotopy groups.

DOI : 10.2140/agt.2016.16.3005
Classification : 14F42, 55T15, 55Q45
Keywords: motivic homotopy theory, stable homotopy group, eta-inverted stable homotopy group, Adams spectral sequence

Guillou, Bertrand  1   ; Isaksen, Daniel  2

1 Department of Mathematics, University of Kentucky, 715 Patterson Office Tower, Lexington, KY 40506-0027, United States
2 Department of Mathematics, Wayne State University, 656 W Kirby, Detroit, MI 48202, United States 
@article{10_2140_agt_2016_16_3005,
     author = {Guillou, Bertrand and Isaksen, Daniel},
     title = {The \ensuremath{\eta}{\textendash}inverted {\ensuremath{\mathbb{R}}{\textendash}motivic} sphere},
     journal = {Algebraic and Geometric Topology},
     pages = {3005--3027},
     year = {2016},
     volume = {16},
     number = {5},
     doi = {10.2140/agt.2016.16.3005},
     url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2016.16.3005/}
}
TY  - JOUR
AU  - Guillou, Bertrand
AU  - Isaksen, Daniel
TI  - The η–inverted ℝ–motivic sphere
JO  - Algebraic and Geometric Topology
PY  - 2016
SP  - 3005
EP  - 3027
VL  - 16
IS  - 5
UR  - http://geodesic.mathdoc.fr/articles/10.2140/agt.2016.16.3005/
DO  - 10.2140/agt.2016.16.3005
ID  - 10_2140_agt_2016_16_3005
ER  - 
%0 Journal Article
%A Guillou, Bertrand
%A Isaksen, Daniel
%T The η–inverted ℝ–motivic sphere
%J Algebraic and Geometric Topology
%D 2016
%P 3005-3027
%V 16
%N 5
%U http://geodesic.mathdoc.fr/articles/10.2140/agt.2016.16.3005/
%R 10.2140/agt.2016.16.3005
%F 10_2140_agt_2016_16_3005
Guillou, Bertrand; Isaksen, Daniel. The η–inverted ℝ–motivic sphere. Algebraic and Geometric Topology, Tome 16 (2016) no. 5, pp. 3005-3027. doi: 10.2140/agt.2016.16.3005

[1] J F Adams, A finiteness theorem in homological algebra, Proc. Cambridge Philos. Soc. 57 (1961) 31 | DOI

[2] M J Andrews, The v1–periodic part of the Adams spectral sequence at an odd prime, PhD thesis, Massachusetts Institute of Technology (2015)

[3] M Andrews, H Miller, Inverting the Hopf map, preprint (2014)

[4] D Dugger, D C Isaksen, Low dimensional Milnor–Witt stems over R, preprint (2015)

[5] B J Guillou, D C Isaksen, The η–local motivic sphere, J. Pure Appl. Algebra 219 (2015) 4728 | DOI

[6] M A Hill, Ext and the motivic Steenrod algebra over R, J. Pure Appl. Algebra 215 (2011) 715 | DOI

[7] D C Isaksen, Stable stems, preprint (2014)

[8] J P May, Matric Massey products, J. Algebra 12 (1969) 533 | DOI

[9] F Morel, On the motivic π0 of the sphere spectrum, from: "Axiomatic, enriched and motivic homotopy theory" (editor J P C Greenlees), NATO Sci. Ser. II Math. Phys. Chem. 131, Kluwer (2004) 219 | DOI

[10] R M F Moss, Secondary compositions and the Adams spectral sequence, Math. Z. 115 (1970) 283 | DOI

Cité par Sources :