[Math] Examples and importance of Embedding (and Non-Embedding) Theorems

Question

An embedding is an injective map into a universal, simpler model object. Many embedding theorems are without obstruction, in the sense that every object which you wish to embed can be embedded. Examples of such theorems are Yoneda lemma, algebraic closure of fields, Nash embedding theorem for Riemannian manifolds. unconditional.

I'm interested in embedding theorems with obstruction. Do you have examples of theorems that give an obstruction to embedding? In the case where there is an obstruction, would you consider the obstruction to be local or instrinsic? Most embedding problems are possible locally, but there is often a local-global obstruction.

The example that led me to this question is Kodiara's embedding theorem that gives an obstruction for a complex manifold to be a submanifold of complex projective space. Here the obstruction is that the manifold must carry a positive line bundle. Positivity of curvature is a local criterion.

PS. Sorry, but I really don't know how to tag this question.

Answer 1

I'm not sure whether you look for such an answer, because it comes from analysis. Analysts use various functional spaces, especially the Sobolev spaces. $W^{s,p}(\Omega;\mathbb R)$ is, roughly speaking, the set of functions with $s$ derivatives in $L^p(\Omega)$ (but $s\ge0$ needs not be an integer).

Sobolev embedding. If $\Omega$ is an open subset with a smooth boundary, and if $\frac1q=\frac1p-\frac{s}{n}$ with $1\le p< q<\infty$, then $W^{s,p}(\Omega;\mathbb R)$ embeds into $L^q(\Omega)$. If instead $sp>n$, then $W^{s,p}(\Omega;\mathbb R)$ embeds into ${\mathcal C}^\alpha(\bar\Omega)$ where $\alpha:=s-\frac{n}{p}$, unless this exponent is an integer.

When the target $\mathbb R$ is replaced by a manifold, the situation may not be so nice. Embedding theorems are related to norm inequalities, which are usually proved first for ${\mathcal C}^\infty$-fields, then extended by means of density of ${\mathcal C}^\infty$ in $W^{s,p}$.

Obstruction (Bethuel 1991). Assume that $p< n$, and let $N$ be a compact manifold of dimension $k$. Then ${\mathcal C}^\infty(\Omega,N)$ is dense in $W^{1,p}(\Omega;N)$ if and only if $\pi_{[p]}(N)=0$, where $[p]$ is the largest integer $\le p$.

The consequence of this is that in some situations, there is a discrepency between $W^{s,p}$ and the closure of ${\mathcal C}^\infty$ under the $W^{s,p}$-norm.