High Quality Content by WIKIPEDIA articles! In differential topology, the Whitney immersion theorem states that for m > 1, any smooth m-dimensional manifold can be immersed in Euclidean 2m ? 1-space. Equivalently, every smooth m-dimensional manifold can be immersed in the 2m ? 1-dimensional sphere (this removes the m > 1 constraint). The weak version, for 2m, is due to transversality (general position, dimension counting): two m-dimensional manifolds in mathbf{R}^{2m} intersect generically in a 0-dimensional space.Massey went on to prove that every n-dimensional manifold is cobordant to a manifold that immerses in S2n ? a(n) where a(n) is the number of 1's that appear in the binary expansion of n. In the same paper, Massey proved that for every n there is manifold (which happens to be a product of real projective spaces) that does not immerse in S2n ? 1 ? a(n).

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