Editing Whitney embedding theorem (section)

===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/>