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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

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.

motivic homotopy theory, stable homotopy group, eta-inverted stable homotopy group, Adams spectral sequence
Mathematical Subject Classification 2010
Primary: 14F42
Secondary: 55T15, 55Q45
Received: 29 October 2015
Revised: 1 March 2016
Accepted: 29 March 2016
Published: 7 November 2016
Bertrand J Guillou
Department of Mathematics
University of Kentucky
715 Patterson Office Tower
Lexington, KY 40506-0027
United States
Daniel C Isaksen
Department of Mathematics
Wayne State University
656 W Kirby
Detroit, MI 48202
United States