Axial maps with further structure

Abstract

For F = R , C F = {\mathbf {R}},{\mathbf {C}} or H {\mathbf {H}} an F F -axial map is defined to be an axial map R P m × R P m → R P m + k {\mathbf {R}}{P^m} \times {\mathbf {R}}{P^m} \to {\mathbf {R}}{P^{m + k}} equivariant with respect to diagonal and trivial F ∗ {F^{\ast }} -actions. Analogously to the real case, it is shown that C {\mathbf {C}} -axial maps correspond to immersions of C P n {\mathbf {C}}{P^n} in R 2 n + k {{\mathbf {R}}^{2n + k}} while (for F = R F = {\mathbf {R}} and for F = C F = {\mathbf {C}} , k k odd) embeddings induce F F -symmaxial maps. Examples are thereby given of symmaxial maps not induced by embeddings of R P n {\mathbf {R}}{P^n} , and of R {\mathbf {R}} -axial maps which are not C {\mathbf {C}} -axial. Furthermore, the relationships which hold when F = R , C F = {\mathbf {R}},{\mathbf {C}} are no longer valid for F = H F = {\mathbf {H}} .