There is a classical statement that a nonorientable closed surface can’t be embedded in $R^3$. And we can proof a weaker statement by Mayer-Vietoris sequence: a closed nonorientable surface $X$ has no neighborhood which is homeomorphism to a mapping cylinder of $f$, where $f$ is a map from a closed orientable surface to $X$ ($H_1(X)$ will has torsion and leads to contradiction in Mayer-Vietoris sequence).
So I am wondering how to see “embedding” with algebraic point of view like the argument above? Intuitively, a geometry interpretation of embedding is that a surface doesn’t intersect itself in a certain space(a Klein bottle must intersects itself in $R^3$, but can be embedded in $R^4$). But it’s hard to write down this idea in the language of mathematics.
I know in Algebraic Geometry, embedding is defined as a surjective morphism of sheaf of rings, which provides a more manageable way to identify “embedding”. And intersection theory may be also a way to deal with “embedding”.
More theories may be needed to handle this question, and I want to know the ideas behind those theories. Thanks in advance!