Whitney's embedding theorem states that any smooth $n$-manifold $M$ can be smoothly embedded into $\Bbb R^{2n}$. In dimension 1 this is tight: the circle cannot be embedded into $\Bbb R^1$. It is a theorem that if $M^n \hookrightarrow \Bbb R^{n+k}$ is a smooth embedding, and $w(M)$ the total Whitney class of the tangent bundle of $M$, then $(w(M)^{-1})_k = 0$. Now consider dimension $m = 2^n$; there $w(\Bbb{RP}^m) = 1+a+a^m$, and hence $w(\Bbb{RP}^m)^{-1} = 1+a+a^2+\dots + a^{m-1}$. Hence $\Bbb{RP}^m$ cannot be embedded into $\Bbb R^{2m-1}$, proving tightness of the bound in these dimensions.

Question: Is Whitney's embedding theorem tight in all dimensions? (Let's restrict to compact manifolds.)

Note that the related immersion theorem, that there is an immersion $M^n \looparrowright \Bbb R^{2n-1}$, does not provide a tight bound for $n=3$, according to this page on the Manifold Atlas project: every compact $3$-manifold immerses into $\Bbb R^4$.

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    $\begingroup$ More generally, the strongest immersion result is the (now-proven) immersion conjecture, which states that every $n$-manifold can be immersed in $\mathbb{R}^{2n-\alpha(n)}$, where $\alpha(n)$ is the number of $1$'s in the binary expansion of $n$. Perhaps the strongest embedding result has a similar form. $\endgroup$
    – Sal
    Dec 13 '14 at 18:52
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    $\begingroup$ Here is a thread addressing the embedding question on MathOverflow. $\endgroup$
    – Sal
    Dec 13 '14 at 18:54
  • $\begingroup$ @Sal Thanks for the references. That answers my question; if you want to make your comments into an answer I would accept it. (I wonder if $\Bbb R^{2n-\alpha(n)+1}$ is a bound on embeddings; it seems like one should be able to take an immersion and massage it into an embedding one dimension higher, but I haven't yet been able to write down a proof.) $\endgroup$
    – user98602
    Dec 13 '14 at 18:56

As referenced in the comments, this MathOverflow question gives the answer as no: every compact $3$-manifold embeds into $\Bbb R^5$, as proved by Wall. The Immersion theorem gives the optimal result for immersions: every compact $n$-manifold embeds into $\Bbb R^{2n-\alpha(n)}$, where $\alpha(n)$ is the number of $1$s in the $2$-adic expansion of $n$. $\Bbb{RP}^n$ cannot embed into any smaller Euclidean space, as seen by inspecting its Stiefel-Whitney classes.

  • $\begingroup$ "Every compact n-manifold embeds into $\mathbb{R}^{2n - \alpha(n)}$" is this true? It seems like you still need to require that the immersion is injective. $\endgroup$ Apr 4 '20 at 4:08

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