The Grothendieck group of spheres with semilinear actions for a compact Lie group

A semilinear G-sphere means a smooth closed G-manifold such that for every subgroup H, the H-fixed point set is a homotopy sphere or empty. We introduce the Grothendieck group V e ( G ) for a certain family of semilinear G-spheres, where e is an idempotent represented by a quasilinear G-disk of the Burnside ring A ( G ) . In this paper we investigate the structure of V e ( G ) and show that V e ( G ) is isomorphic to JO ( G ) ; the Grothendieck group for linear G-spheres. On the other hand, both V e ( G ) and JO ( G ) are naturally regarded as subgroups of the homotopy representation group V ∞ ( G ) . We also show that V e ( G ) and JO ( G ) are different subgroups of V ∞ ( G ) when e ≠ 1 .