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Immersions and Embeddings Up to Cobordism

Published online by Cambridge University Press:  20 November 2018

Richard L. W. Brown
Richard L. W. Brown*
Affiliation:
York University, Toronto, Ontario
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In 1944 Whitney proved that any differentiable n-manifold (n ≧ 2) can be (differentiably) immersed in R2n–1[15] and embedded in R2n [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 R2n–α(n) and embeds in R2n–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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 6 , December 1971 , pp. 1102 - 1115
Copyright
Copyright © Canadian Mathematical Society 1971

References

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