$\mathbb {Z}/2$-equivariant and $\mathbb {R}$-motivic stable stems
HTML articles powered by AMS MathViewer
- by Daniel Dugger and Daniel C. Isaksen PDF
- Proc. Amer. Math. Soc. 145 (2017), 3617-3627 Request permission
Abstract:
We establish an isomorphism between the stable homotopy groups $\hat {\pi }^{\mathbb {R}}_{s,w}$ of the 2-completed $\mathbb {R}$-motivic sphere spectrum and the stable homotopy groups $\hat {\pi }^{\mathbb {Z}/2}_{s,w}$ of the 2-completed $\mathbb {Z}/2$-equivariant sphere spectrum, valid in the range $s \geq 3 w - 5$ or $s \leq -1$.References
- Shôrô Araki and Kouyemon Iriye, Equivariant stable homotopy groups of spheres with involutions. I, Osaka Math. J. 19 (1982), no. 1, 1–55. MR 656233
- Glen E. Bredon, Equivariant homotopy, Proc. Conf. on Transformation Groups (New Orleans, La., 1967) Springer, New York, 1968, pp. 281–292. MR 0250303
- Daniel Dugger and Daniel C. Isaksen, Low-dimensional Milnor-Witt stems over $\mathbb {R}$ (2015), to appear in Annals $K$-Theory.
- Po Hu and Igor Kriz, Real-oriented homotopy theory and an analogue of the Adams-Novikov spectral sequence, Topology 40 (2001), no. 2, 317–399. MR 1808224, DOI 10.1016/S0040-9383(99)00065-8
- P. Hu, I. Kriz, and K. Ormsby, Convergence of the motivic Adams spectral sequence, J. K-Theory 7 (2011), no. 3, 573–596. MR 2811716, DOI 10.1017/is011003012jkt150
- Kouyemon Iriye, Equivariant stable homotopy groups of spheres with involutions. II, Osaka J. Math. 19 (1982), no. 4, 733–743. MR 687770
- Fabien Morel, An introduction to $\Bbb A^1$-homotopy theory, Contemporary developments in algebraic $K$-theory, ICTP Lect. Notes, XV, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2004, pp. 357–441. MR 2175638
- Fabien Morel, The stable ${\Bbb A}^1$-connectivity theorems, $K$-Theory 35 (2005), no. 1-2, 1–68. MR 2240215, DOI 10.1007/s10977-005-1562-7
- Fabien Morel and Vladimir Voevodsky, $\textbf {A}^1$-homotopy theory of schemes, Inst. Hautes Études Sci. Publ. Math. 90 (1999), 45–143 (2001). MR 1813224
- Markus Szymik, Equivariant stable stems for prime order groups, J. Homotopy Relat. Struct. 2 (2007), no. 1, 141–162. MR 2369156
- Vladimir Voevodsky, Motivic cohomology with $\textbf {Z}/2$-coefficients, Publ. Math. Inst. Hautes Études Sci. 98 (2003), 59–104. MR 2031199, DOI 10.1007/s10240-003-0010-6
- Vladimir Voevodsky, Motivic Eilenberg-MacLane spaces, Publ. Math. Inst. Hautes Études Sci. 112 (2010), 1–99. MR 2737977, DOI 10.1007/s10240-010-0024-9
Additional Information
- Daniel Dugger
- Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403
- MR Author ID: 665595
- Email: ddugger@math.uoregon.edu
- Daniel C. Isaksen
- Affiliation: Department of Mathematics, Wayne State University, Detroit, Michigan 48202
- MR Author ID: 611825
- Email: isaksen@wayne.edu
- Received by editor(s): March 30, 2016
- Received by editor(s) in revised form: September 18, 2016
- Published electronically: February 21, 2017
- Communicated by: Michael A. Mandell
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 3617-3627
- MSC (2010): Primary 55Q10; Secondary 55Q91
- DOI: https://doi.org/10.1090/proc/13505
- MathSciNet review: 3652813