Yiğit, Uğur \(C_2\)-equivariant James splitting and \(C_2\)-EHP sequences. (English) Zbl 1461.55010 Topology Appl. 289, Article ID 107485, 13 p. (2021). Let \(C_2\) denote the cyclic group of order \(2\) and write \(\sigma\) for the sign representation and \(\rho=1+\sigma\) for the regular representation. The author’s first result is the \(C_2\)-equivariant generalization of the James splitting: for a \(C_2\)-representations \(V\) of dimension \(|V|\), there is a \(C_2\)-weak homotopy equivalence \[ \Sigma^\sigma \Omega^\sigma \Sigma^\sigma S^V \simeq \bigvee_{k=1}^\infty S^{|V|k+\sigma}. \] The second result of the paper is the construction of a \(C_2\)-equivariant EHP sequence \[ S^V \xrightarrow{E^\sigma} \Omega^\sigma S^{V+\sigma} \xrightarrow{H^\sigma} \Omega^\sigma S^{V\otimes \rho+ \sigma} \] which provides a tool for computing unstable \(C_2\)-homotopy groups of equivariant spheres. Here \(H^\sigma\) is the Hopf invariant map and \(E^\sigma\) is the suspension (Einhängung) map. Reviewer: Luca Pol (Regensburg) MSC: 55P91 Equivariant homotopy theory in algebraic topology 55Q91 Equivariant homotopy groups 55Q45 Stable homotopy of spheres 55T99 Spectral sequences in algebraic topology Keywords:equivariant homotopy; equivariant EHP sequence; equivariant James splitting PDFBibTeX XMLCite \textit{U. Yiğit}, Topology Appl. 289, Article ID 107485, 13 p. (2021; Zbl 1461.55010) Full Text: DOI References: [1] Araki, Shôrô; Iriye, Kouyemon, Equivariant stable homotopy groups of spheres with involutions. I, Osaka J. Math., 19, 1, 1-55 (1982) · Zbl 0488.55012 [2] Araki, Shôrô; Murayama, Mitutaka, τ-cohomology theories, Jpn. J. Math. New Ser., 4, 2, 363-416 (1978) · Zbl 0411.55004 [3] Bredon, Glen E., Equivariant stable stems, Bull. Am. Math. Soc., 73, 269-273 (1967) · Zbl 0152.21803 [4] Bredon, Glen E., Equivariant homotopy, (Proc. Conf. on Transformation Groups. Proc. Conf. on Transformation Groups, New Orleans, La., 1967 (1968), Springer: Springer New York), 281-292 · Zbl 0162.27202 [5] Freudenthal, H., Über die Klassen der Sphärenabbildungen I. Grosse Dimnesionionen, Compos. Math., 5, 299-314 (1938) · JFM 63.1161.02 [6] Hauschild, H., Äquivariante Homotopie. I, Arch. Math. (Basel), 29, 2, 158-165 (1977) · Zbl 0367.55013 [7] Hill, Michael A., On the algebras over equivariant little disks (2017) [8] Landweber, Peter S., On equivariant maps between spheres with involutions, Ann. Math. (2), 89, 125-137 (1969) · Zbl 0174.26204 [9] Lewis, L. Gaunce, The equivariant Hurewicz map, Trans. Am. Math. Soc., 329, 2, 433-472 (1992) · Zbl 0769.54042 [10] Rybicki, Sławomir, \( Z_2\)-equivariant James construction, Bull. Pol. Acad. Sci., Math., 39, 1-2, 83-90 (1991) · Zbl 0763.55009 [11] Serre, Jean-Pierre, Homologie singulière des espaces fibrés. Applications, Ann. Math. (2), 54, 425-505 (1951) · Zbl 0045.26003 [12] Whitehead, G. W., Elements of Homotopy Theory (1978), Springer-Verlag: Springer-Verlag New York · Zbl 0406.55001 [13] Xicoténcatl, Miguel A., On \(\mathbb{Z}_2\)-equivariant loop spaces, (Recent Developments in Algebraic Topology. Recent Developments in Algebraic Topology, Contemp. Math., vol. 407 (2006), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 183-191 · Zbl 1104.55004 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.