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Some remarks on projective Stiefel manifolds, immersions of projective spaces and spheres


Author: Larry Smith
Journal: Proc. Amer. Math. Soc. 80 (1980), 663-669
MSC: Primary 57R42
DOI: https://doi.org/10.1090/S0002-9939-1980-0587951-X
MathSciNet review: 587951
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Abstract: Let $ {M^m}$ be a closed smooth manifold, $ M[unk]{{\mathbf{R}}^{2m}}$ an immersion and $ {\tilde M^m}{ \downarrow ^\pi }{M^m}$ a double covering. For m odd we show that the normal bundle $ \tilde \nu \downarrow \tilde M$ of the immersion $ \tilde M{ \downarrow ^\pi }M[unk]{{\mathbf{R}}^{2m}}$ is independent of $ \varphi $ and applying this to $ M = {\mathbf{R}}P(m)$ reobtain the result of E. H. Brown that a symmetric immersion $ {S^m}[unk]{{\mathbf{R}}^{2m}}$ is regularly homotopic to an embedding iff $ m = {2^p} - 1$.


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DOI: https://doi.org/10.1090/S0002-9939-1980-0587951-X
Article copyright: © Copyright 1980 American Mathematical Society

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