Immersions of $n$-manifolds in $\mathbb{R}^n$ versus embeddings in $\mathbb{R}^{n+1}$

Question

Let $M$ be a connected noncompact parallelisable smooth $n$-manifold. (By Hirsch-Smale theory, $M$ can be immersed into $\mathbb{R}^n$.) I am interested in the following two properties:

P1: $M$ can be smoothly embedded into $\mathbb{R}^{n+1}$.

P2: $M$ admits a smooth embedding into $\mathbb{R}^{n+1}$ which is transverse to the constant vector field $e_{n+1}$ on $\mathbb{R}^{n+1}$.

Questions/problems (my main concern is Q2):

Q1: Are P1 and P2 equivalent? If not, then I am asking Q2 and Q3 also for P1, but I am primarily interested in P2.

Q2: I'd like to have an example $M$ which does not have property P2. I expect that one exists. If not, then I'd like to have a proof of the nonexistence.

Q3: Assuming that an $M$ without P2 exists: what is the smallest dimension $n$ where P2-violation is possible?

Answer 1

I was slightly confused by the wording of your question. I interpret it that you are asking first if an immersion into $R^n$ implies an embedding into $R^{n+1}$ (P1) and if so, if you get an embedding that is transverse to a coordinate vector field (P2). It wasn't clear to me whether you wanted to specify the immersion in $R^n$ to which the embedding resulting from P2 would project.

It is not true that parallelizability implies embedding into $R^{n+1}$, so P1 and a fortiori, P2 do not follow from the existence of an immersion in $R^n$. The easiest examples of this are for $n=3$, where you can take $M$ to be a punctured lens space $L(2p,q) - pt$, where the linking form (on the torsion in the homology) provides an obstruction. This goes back quite some time to W. Hantzsche: Einlagerung von Mannigfeltigkeiten in euklidishe Raume. Math. Zeit. 43 (1938). 38-58 and D. B. A. Epstein: Embedding punctured manifolds. Proc. A.M.S. 16 (1965), 175-176. A more modern treatment of the 3-dimensional case is in the paper On embeddings of 3-manifolds in 4-space, by P. Gilmer and C. Livingston, Topology Vol. 22. No. 3, pp. 241-252, 1983. There are examples in higher dimensions as well.

With regard to the equivalence of P1 and P2, I don't know the answer. An example of Carter and Saito (Surfaces in 3-space that do not lift to embeddings in 4-space. Knot theory (Warsaw, 1995), 29–47, Banach Center Publ., 42, Polish Acad. Sci., Warsaw, 1998) seems somewhat relevant, although they are concerned with closed surfaces, rather than the specified non-compact manifold. A potential counterexample to P1 implies P2 might be found amongst higher dimensional lens spaces, where stable parallelizability of a lens space whose fundamental group has prime-power order implies that the punctured manifold embeds in codimension one (so P1 holds). (This is according to an old paper of mine: Imbedding punctured lens spaces and connected sums. Pacific J. Math. 113 (1984), no. 2, 481–491.) But I don't know if there would be an obstruction to embedding transverse to a given vector field.