Geodesic
Immersions and Embeddings Up to Cobordism
Canadian journal of mathematics, Tome 23 (1971) no. 6, pp. 1102-1115

Voir la notice de l'article provenant de la source Cambridge University Press

In 1944 Whitney proved that any differentiable n-manifold (n ≧ 2) can be (differentiably) immersed in R 2n–1[15] and embedded in R 2n [14]. Whitney's results are best possible when n = 2r. (One uses a simple argument involving the dual Stiefel-Whitney classes of real projective space Pn . See [9, pp. 14, 15].) However, there is a widely known conjecture that any R-manifold (n ≧ 2) immerses in R 2n–α(n) and embeds in R 2n–α(n)+1. Here, α(n) denotes the number of ones in the binary expansion of n. We prove (Theorem 5.1) that every compact manifold is cobordant to a manifold that immerses in (2n – α(n))-space and embeds in (2n – α(n) + 1)-space. (See § 1 for the definition of cobordant manifolds.) It is well known that if the conjecture were true it would be the best possible result. 
Brown, Richard L. W. Immersions and Embeddings Up to Cobordism. Canadian journal of mathematics, Tome 23 (1971) no. 6, pp. 1102-1115. doi: 10.4153/CJM-1971-116-8
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