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
Famous quotes containing the words sharper and/or results:
“The greater the decrease in the social significance of an art form, the sharper the distinction between criticism and enjoyment by the public. The conventional is uncritically enjoyed, and the truly new is criticized with aversion.”
—Walter Benjamin (1892–1940)
“Pain itself can be pleasurable accidentally in so far as it is accompanied by wonder, as in stage-plays; or in so far as it recalls a beloved object to one’s memory, and makes one feel one’s love for the thing, whose absence gives us pain. Consequently, since love is pleasant, both pain and whatever else results from love, in so far as they remind us of our love, are pleasant.”
—Thomas Aquinas (c. 1225–1274)