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\(C_2\)-equivariant and \(\mathbb{R}\)-motivic stable stems. II. (English) Zbl 1479.55022

Just as the computation of the stable stems is one of the fundamental questions in algebraic topology, the homotopy groups of the \(C_2\)-equivariant and \(\mathbb{R}\)-motivic stable stems are fundamental to \(C_2\)-equivariant and \(\mathbb{R}\)-motivic homotopy theory. These computations also give additional information about the classical stable stems and therefore global results about the \(\mathbb{R}\)-motivic stable stems and the \(C_2\)-equivariant ones are certainly important.
The paper under review provides a range of degrees, given by an inequality involving the stem and co-weight, in which the \(C_2\)-equivariant stable stems and the \(\mathbb{R}\)-motivic stable stems agree. This work extends a previous result of D. Dugger and D. C. Isaksen [Proc. Am. Math. Soc. 145, No. 8, 3617–3627 (2017; Zbl 1421.55011)] providing an isomorphism between the \(C_2\)-equivariant stable stems and \(\mathbb{R}\)-motivic stable stems in a smaller range. This result is used in a key way to study Mahowald invariants by Eva Belmont and Daniel C. Isaksen. The result is useful from a computational perspective and it is interesting in its own right because it provides qualitative information about the difference between the \(\mathbb{R}\)-motivic and \(C_2\)-equivariant stable stems. The paper concludes by describing why the main theorem is sharp in a sense the authors make precise.
The proof improves on the method of Daniel Dugger and Daniel C. Isaksen using the respective cobar complexes by instead applying \(\mathbb{R}\)-motivic and \(C_2\)-equivariant \(\rho\)-Bockstein spectral sequences. This removes some of the extra data that appears in the cobar complexes, but does not survive to the \(E_2\)-pages of the respective Adams spectral sequences. The proof is therefore also interesting in its own right as an example of how the \(\rho\)-Bockstein spectral sequence can simplify computations of \(\mathbb{R}\)-motivic and \(C_2\)-equivariant homotopy groups.

MSC:

55Q10 Stable homotopy groups
55Q91 Equivariant homotopy groups
14F42 Motivic cohomology; motivic homotopy theory
55Q45 Stable homotopy of spheres
55T15 Adams spectral sequences

Citations:

Zbl 1421.55011
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Full Text: DOI arXiv

References:

[1] Belmont, Eva; Isaksen, Daniel C., \( \mathbb{R} \)-motivic stable stems (2020) · Zbl 1479.55022
[2] Dugger, Daniel; Isaksen, Daniel C., The motivic Adams spectral sequence, Geom. Topol., 14, 2, 967-1014 (2010) · Zbl 1206.14041
[3] Dugger, Daniel; Isaksen, Daniel C., Low-dimensional Milnor-Witt stems over \(\mathbb{R} \), Ann. K-Theory, 2, 2, 175-210 (2017) · Zbl 1400.14064
[4] Dugger, Daniel; Isaksen, Daniel C., \( \mathbb{Z}/2\)-equivariant and \(\mathbb{R} \)-motivic stable stems, Proc. Amer. Math. Soc., 145, 8, 3617-3627 (2017) · Zbl 1421.55011
[5] Gheorghe, Bogdan; Wang, Guozhen; Xu, Zhouli, The special fiber of the motivic deformation of the stable homotopy category is algebraic, Acta. Math.
[6] Guillou, Bertrand J.; Hill, Michael A.; Isaksen, Daniel C.; Ravenel, Douglas Conner, The cohomology of \(C_2\)-equivariant \(\mathcal{A}(1)\) and the homotopy of \(\text{ko}_{C_2} \), Tunis. J. Math., 2, 3, 567-632 (2020) · Zbl 1440.14124
[7] Guillou, Bertrand J.; Isaksen, Daniel C., The Bredon-Landweber region in \(C_2\)-equivariant stable homotopy groups · Zbl 1453.55012
[8] Heller, J.; Ormsby, K., Galois equivariance and stable motivic homotopy theory, Trans. Amer. Math. Soc., 368, 11, 8047-8077 (2016) · Zbl 1346.14049
[9] Isaksen, Daniel C., Stable stems, Mem. Amer. Math. Soc., 262, 1269, viii+159 pp. (2019) · Zbl 1454.55001
[10] Isaksen, Daniel C.; Wang, Guozhen; Xu, Zhouli, More stable stems (2020)
[11] Miller, Haynes Robert, Some algebraic aspects of the Adams-Novikov spectral sequence, 103 pp. (1975), ProQuest LLC, Ann Arbor, MI
[12] Novikov, S. P., Methods of algebraic topology from the point of view of cobordism theory, Izv. Akad. Nauk SSSR Ser. Mat., 31, 855-951 (1967)
[13] Morel, Fabien; Voevodsky, Vladimir, \( \mathbf{A}^1\)-homotopy theory of schemes, Inst. Hautes \'{E}tudes Sci. Publ. Math., 90, 45-143 (2001) (1999) · Zbl 0983.14007
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