Sharper Results
Although every -manifold embeds in, one can frequently do better. Let denote the smallest integer so that all compact connected -manifolds embed in . Whitney's strong embedding theorem states that . For we have, as the circle and the Klein bottle show. More generally, for we have, as the -dimensional real projective space show. Whitney's result can be improved to unless is a power of 2. This is a result of Haefliger–Hirsch and C.T.C. Wall ; these authors used important preliminary results and particular cases proved by M. Hirsch, W. Massey, S. Novikov and V. Rokhlin, see section 2 of this survey. At present the function is not known in closed-form for all integers (compare to the Whitney immersion theorem, where the analogous number is known).
Read more about this topic: Whitney Embedding Theorem
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