1
$\begingroup$

I know that for smooth embedding a manifold $M^n$ into $\mathbb R^N$, the Whitney embedding theorem says it's possible if $N\geq2n$. Moreover this codimension is tight (at least as a linear function of $n$) thanks to $\mathbb{RP}^n$ for certain $n$. My question is whether you can improve the codimension if $M$ is say compact and simply connected. For example, in the case $n=1,2$ we can take $N=n+1$.

$\endgroup$

1 Answer 1

1
$\begingroup$

Every closed orientable (smooth) n-manifold (smoothly) embeds in $\mathbb{R^{2n−1}}$. This is true for $n>4$ by Haefliger-Hirsch, $n=3$ by C.T.C. Wall.(which says that every closed 3 manifold canbe embedded in $\mathbb R^5$).

And we know that every simply connected manifold is orientable. So it gives us a better bound as you desired.

Observe that when $n=3$ any simply connected closed manifold is actually $S^3$ (Poincare conjecture). So in n=3, you can choose $N=4$. But For $n>4$ it is not true. For example take $\mathbb{CP^2}$. {Since it's signature is non-zero so it cannot be a boundary of a orientable $5$ manifold.}

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .