Oriented \(Z_ 4\) actions without stationary points
This paper studies oriented cobordism groups with actions of the groups \(\mathbb{Z}/4\) and \(\mathbb{Z}/2\), in particular, the restriction map \(r : \Omega_*(\mathbb{Z}/4;p) \to \Omega_*(\mathbb{Z}/2; \text{All})\) relating fixed point free \(\mathbb{Z}/4\)-actions to their restrictions to \(\mathbb{Z}/2\). The image of \(r\) is shown to lie in a certain homology theory obtained from differentials on the relative Wall cobordism groups \(W_*(\mathbb{Z}/2\); All, Free) [cf. e.g. \textit{C. Kosniowski, E. Ossa}, Proc. Lond. Math. Soc., III. Ser. 44, 267-290 (1982; Zbl 0437.57017)], which is calculated subsequently. The kernel fo \(r\) is identified in terms of two types of extensions of the torsion free part of \(\Omega_*(\mathbb{Z}/2;\) All) and of the torsion part of order 2 in \(\Omega_*(\mathbb{Z}/4;p)\) studied by the author in an earlier paper. Examples arising from standard actions on complex projective spaces illustrate the theory.
scientific article; zbMATH DE number 621958
scientific article