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Induction on symmetric axial maps and embeddings of projective spaces


Author: A. J. Berrick
Journal: Proc. Amer. Math. Soc. 60 (1976), 276-278
MSC: Primary 57D40
DOI: https://doi.org/10.1090/S0002-9939-1976-0420661-7
MathSciNet review: 0420661
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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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0420661-7
Keywords: Axial map, embedding, immersion, isotopy, projective space, regular homotopy, Smale invariant, Stiefel manifold
Article copyright: © Copyright 1976 American Mathematical Society

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