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

Abstract

Let M m {M^m} be a closed smooth manifold, M [ u n k ] R 2 m M[unk]{{\mathbf {R}}^{2m}} an immersion and M ~ m ↓ π M m {\tilde M^m}{ \downarrow ^\pi }{M^m} a double covering. For m odd we show that the normal bundle ν ~ ↓ M ~ \tilde \nu \downarrow \tilde M of the immersion M ~ ↓ π M [ u n k ] R 2 m \tilde M{ \downarrow ^\pi }M[unk]{{\mathbf {R}}^{2m}} is independent of φ \varphi and applying this to M = R P ( m ) M = {\mathbf {R}}P(m) reobtain the result of E. H. Brown that a symmetric immersion S m [ u n k ] R 2 m {S^m}[unk]{{\mathbf {R}}^{2m}} is regularly homotopic to an embedding iff m = 2 p − 1 m = {2^p} - 1 .