Volume 16, issue 5 (2016)

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

Bertrand J Guillou and Daniel C Isaksen

Algebraic & Geometric Topology 16 (2016) 3005–3027
Abstract

We use an Adams spectral sequence to calculate the $ℝ$–motivic stable homotopy groups after inverting $\eta$. The first step is to apply a Bockstein spectral sequence in order to obtain ${h}_{1}$–inverted $ℝ$–motivic $Ext$ groups, which serve as the input to the $\eta$–inverted $ℝ$–motivic Adams spectral sequence. The second step is to analyze Adams differentials. The final answer is that the Milnor–Witt $\left(4k-1\right)$–stem has order ${2}^{u+1}$, 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 $\eta$–inverted $ℝ$–motivic stable homotopy groups.

Keywords
motivic homotopy theory, stable homotopy group, eta-inverted stable homotopy group, Adams spectral sequence
Mathematical Subject Classification 2010
Primary: 14F42
Secondary: 55T15, 55Q45