# Topological embedding of Klein bottle into $\mathbb{R}^4$ that projects to usual “beer-bottle” surface in $\mathbb{R}^3$?

What is an explicit topological embedding of the Klein bottle into $$\mathbb{R}^4$$ whose projection, of some sort, down to $$\mathbb{R}^3$$ gives the usual "beer-bottle" immersed surface (https://upload.wikimedia.org/wikipedia/commons/8/8a/Surface_of_Klein_bottle_with_traced_line.svg), and then how does that embedding and the projection yield an explicit "topological immersion" of the Klein bottle into $$\mathbb{R}^3$$?

By a "topological immersion" I mean at least a continuous local homeomorphism.

What I’m looking for, more precisely, is a map $$f: [a, b] \times [c, d] \rightarrow \mathbb{R}^4$$ with the following properties:

• $$[a, b] = [c, d] = [0, 1]$$, or perhaps more conveniently, $$[a, b] = [c, d] = [0, 2 \pi]$$;
• $$f$$ is constant on equivalence classes under the usual equivalence relation on the rectangle that identifies horizontal edges going in the same direction but identifies vertical edges in opposite directions;
• by passing to the quotient, $$f$$ induces an embedding $$f^{\ast}$$ of the Klein bottle $$K$$ into $$\mathbb{R}^4$$;
• the composite $$g = p \circ f^{\ast}$$ is an immersion of $$K$$ into $$\mathbb{R}^3$$ whose image is that “beer-bottle” surface $$S$$, where the "projection" $$p$$ has a form such as $$p(x_1, x_2, x_3, x_4) = (x_1, x_2, x_3 \cos \alpha + x_4 \sin \alpha)$$; and
• the composite $$q \circ g$$ is therefore a parameterization of that surface $$S$$, where $$q : [a, b] \times [c, d] \to K$$ is the quotient map.
• I guess that the usual schematic “embedding” $f$ of the Klein bottle into $\Bbb R^3$ is not a local homeomorphism (for instance, if $f(x)$ is a crossing point then I guess that for a sufficiently small neighborhood $U$ of $x$ the image $f(U)$ would be non-open, contradicting to a definition of a local homeomorphism). – Alex Ravsky Aug 27 '19 at 15:44