## Algebraic & Geometric Topology

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

#### Abstract

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.

#### Article information

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

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

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

#### 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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