Algebraic & Geometric Topology

The $\eta$–inverted $\mathbb{R}$–motivic sphere

Bertrand Guillou and Daniel Isaksen

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Abstract

We use an Adams spectral sequence to calculate the –motivic stable homotopy groups after inverting η. The first step is to apply a Bockstein spectral sequence in order to obtain h1 –inverted –motivic Ext groups, which serve as the input to the η–inverted –motivic Adams spectral sequence. The second step is to analyze Adams differentials. The final answer is that the Milnor–Witt (4k1)–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 –motivic stable homotopy groups.

Article information

Source
Algebr. Geom. Topol., Volume 16, Number 5 (2016), 3005-3027.

Dates
Received: 29 October 2015
Revised: 1 March 2016
Accepted: 29 March 2016
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1510841237

Digital Object Identifier
doi:10.2140/agt.2016.16.3005

Mathematical Reviews number (MathSciNet)
MR3572357

Zentralblatt MATH identifier
06653767

Subjects
Primary: 14F42: Motivic cohomology; motivic homotopy theory [See also 19E15]
Secondary: 55T15: Adams spectral sequences 55Q45: Stable homotopy of spheres

Keywords
motivic homotopy theory stable homotopy group eta-inverted stable homotopy group Adams spectral sequence

Citation

Guillou, Bertrand; Isaksen, Daniel. The $\eta$–inverted $\mathbb{R}$–motivic sphere. Algebr. Geom. Topol. 16 (2016), no. 5, 3005--3027. doi:10.2140/agt.2016.16.3005. https://projecteuclid.org/euclid.agt/1510841237


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References

  • J,F Adams, A finiteness theorem in homological algebra, Proc. Cambridge Philos. Soc. 57 (1961) 31–36
  • M,J Andrews, The $v\sb 1$–periodic part of the Adams spectral sequence at an odd prime, PhD thesis, Massachusetts Institute of Technology (2015) Available at \setbox0\makeatletter\@url http://search.proquest.com/docview/1720302322 {\unhbox0
  • M Andrews, H Miller, Inverting the Hopf map, preprint (2014) Available at \setbox0\makeatletter\@url http://www-math.mit.edu/~hrm/papers/andrews-miller-jul08.pdf {\unhbox0
  • D Dugger, D,C Isaksen, Low dimensional Milnor–Witt stems over ${\mathbb R}$, preprint (2015)
  • B,J Guillou, D,C Isaksen, The $\eta$–local motivic sphere, J. Pure Appl. Algebra 219 (2015) 4728–4756
  • M,A Hill, Ext and the motivic Steenrod algebra over $\mathbb R$, J. Pure Appl. Algebra 215 (2011) 715–727
  • D,C Isaksen, Stable stems, preprint (2014)
  • J,P May, Matric Massey products, J. Algebra 12 (1969) 533–568
  • F Morel, On the motivic $\pi\sb 0$ of the sphere spectrum, from “Axiomatic, enriched and motivic homotopy theory” (J,P,C Greenlees, editor), NATO Sci. Ser. II Math. Phys. Chem. 131, Kluwer, Dordrecht (2004) 219–260
  • R,M,F Moss, Secondary compositions and the Adams spectral sequence, Math. Z. 115 (1970) 283–310