Editing Whitney embedding theorem (section)

==Sharper results==
Although every {{mvar|n}}-manifold embeds in {{tmath|\R^{2n},}} one can frequently do better.  Let {{math|''e''(''n'')}} denote the smallest integer so that all compact connected {{mvar|n}}-manifolds embed in {{tmath|\R^{e(n)}.}} Whitney's strong embedding theorem states that {{math|''e''(''n'') ≤ 2''n''}}.  For {{math|1=''n'' = 1, 2}} we have {{math|1=''e''(''n'') = 2''n''}}, as the [[circle]] and the [[Klein bottle]] show. More generally, for {{math|1=''n'' = 2<sup>''k''</sup>}} we have {{math|1=''e''(''n'') = 2''n''}}, as the {{math|2<sup>''k''</sup>}}-dimensional [[real projective space]] show. Whitney's result can be improved to {{math|''e''(''n'') ≤ 2''n'' − 1}} unless {{mvar|n}} is a power of 2. This is a result of [[André Haefliger]] and [[Morris Hirsch]] (for {{math|''n'' > 4}}) and [[C. T. C. Wall]] (for {{math|1=''n'' = 3}}); these authors used important preliminary results and particular cases proved by Hirsch, [[William S. Massey]], [[Sergei Novikov (mathematician)|Sergey Novikov]] and [[Vladimir Rokhlin (Soviet mathematician)|Vladimir Rokhlin]].<ref name=skopenkov2>See section 2 of Skopenkov (2008)</ref> At present the function {{mvar|e}} is not known in closed-form for all integers (compare to the [[Whitney immersion theorem]], where the analogous number is known).

===Restrictions on manifolds===
One can strengthen the results by putting additional restrictions on the manifold. For example, the [[n-sphere|{{mvar|n}}-sphere]] always embeds in {{tmath|\R^{n+1} }}&nbsp;– which is the best possible (closed {{mvar|n}}-manifolds cannot embed in {{tmath|\R^n}}). Any compact ''orientable'' surface and any compact surface ''with non-empty boundary'' embeds in {{tmath|\R^3,}} though any ''closed non-orientable'' surface needs {{tmath|\R^4.}}

If {{mvar|N}} is a compact orientable {{mvar|n}}-dimensional  manifold, then {{mvar|N}} embeds in {{tmath|\R^{2n-1} }} (for {{mvar|n}} not a power of 2 the orientability condition is superfluous). For {{mvar|n}} a power of 2 this is a result of [[André Haefliger]] and [[Morris Hirsch]] (for {{math|''n'' > 4}}), and Fuquan Fang (for {{math|1=''n'' = 4}}); these authors used important preliminary results proved by Jacques Boéchat and Haefliger, [[Simon Donaldson]], Hirsch and [[William S. Massey]].<ref name=skopenkov2/> Haefliger proved that if {{mvar|N}} is a compact {{mvar|n}}-dimensional [[n-connected|{{mvar|k}}-connected]] manifold, then {{mvar|N}} embeds in {{tmath|\R^{2n-k} }} provided {{math|2''k'' + 3 ≤ ''n''}}.<ref name=skopenkov2/>