The η–inverted ℝ–motivic sphere

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 $h_{1}$ –inverted $ℝ$ –motivic $E x t$ 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 $(4 k - 1)$ –stem has order $2^{u + 1}$ , where $u$ is the $2$ –adic valuation of $4 k$ . 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.

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Keywords

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

Mathematical Subject Classification 2010

Primary: 14F42

Secondary: 55T15, 55Q45

References

Publication

Received: 29 October 2015

Revised: 1 March 2016

Accepted: 29 March 2016

Published: 7 November 2016

Authors

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

Volume 16, issue 5 (2016)

Bertrand J Guillou and Daniel C Isaksen

Abstract