A note on the existence of 𝐺-maps between spheres

Waner, Stefan

doi:10.1090/S0002-9939-1987-0866449-1

A note on the existence of $G$-maps between spheres
HTML articles powered by AMS MathViewer

by Stefan Waner PDF

Proc. Amer. Math. Soc. 99 (1987), 179-181 Request permission

Abstract:

Let $G$ be a finite group, and let $V$ and $W$ be finite-dimensional real orthogonal $G$-modules with $V \supset W$, and with unit spheres $S(V)$ and $S(W)$ respectively. The purpose of this note is to give necessary sufficient conditions for the existence of a $G$-map $J:S(V) \to S(W)$ in terms of the Burnside ring of $G$ and its relationship with $V$ and $W$. Note that if $W$ has a nonzero fixed point, such a $G$-map always exists, so for nontriviality, we assume this not the case.

References

Tammo tom Dieck, Transformation groups and representation theory, Lecture Notes in Mathematics, vol. 766, Springer, Berlin, 1979. MR 551743, DOI 10.1007/BFb0085965
G. B. Segal, Equivariant stable homotopy theory, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 59–63. MR 0423340
Stefan Waner and Yihren Wu, The local structure of tangent $G$-vector fields, Topology Appl. 23 (1986), no. 2, 129–143. MR 855452, DOI 10.1016/0166-8641(86)90034-9