Axial maps with further structure

Berrick, A. J.

doi:10.1090/S0002-9939-1976-0397750-9

Axial maps with further structure
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by A. J. Berrick PDF

Proc. Amer. Math. Soc. 54 (1976), 413-416 Request permission

Abstract:

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 Furthermore, the relationships which hold when $F = {\mathbf {R}},{\mathbf {C}}$ are no longer valid for $F = {\mathbf {H}}$.

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Additional Information

Journal: Proc. Amer. Math. Soc. 54 (1976), 413-416
DOI: https://doi.org/10.1090/S0002-9939-1976-0397750-9
MathSciNet review: 0397750