Algebraic & Geometric Topology

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

Bertrand Guillou and Daniel Isaksen

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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

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

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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