Induction on symmetric axial maps and embeddings of projective spaces

Abstract:

A homotopy class of axial maps ${P^n} \times {P^n} \to {P^{n + k}}$ determines an invariant in ${\pi _n}({V_{n + k + 1,n + 1}})\;(2k \geqslant n + 2)$. If an axial map is symmetric and has trivial invariant it extends to a symmetric axial map ${P^{n + 1}} \times {P^{n + 1}} \to {P^{n + k + 1}}$. An immersion of ${P^n}$ in ${R^{n + k}}$ lifts to an immersion of ${S^n}$ in ${R^{n + k}}$ and so has a Smale invariant. For $j:{R^{n + k}}\hookrightarrow {R^{n + k + 2}},2k \geqslant n + 2$ (resp. $2k \geqslant n + 3$), any embedding $a:{P^n} \to {R^{n + k}}$ with trivial Smale invariant induces an embedding of ${P^{n + 1}}$ in ${R^{n + k + 2}}$ whose restriction to ${P^n}$ is regularly homotopic (resp. isotopic) to ja.