I wonder if someone can clarify whether it is known that every closed, orientable surface (2-manifold) has a smooth isometric immersion in $\mathbb{R}^4$?

This topic has been discussed rather thoroughly in the earlier MO question, "Nash embedding theorem for 2D manifolds." However, I am puzzled by two claims that seem at odds, but are surely not—rather I just don't understand the terminology.

First, Gromov established that every compact surface
has a smooth embedding in $\mathbb{R}^5$.
This is from his 1986 book,
*Partial Differential Relations*.
See the above MO question for a quote.

Second, as user jc said,
"Poznyak proved that $\mathbb{R}^4$ works
for any compact part of a complete surface."
Here is a more precise statement of Poznyak's
result, from the book
by Jia-Xing Hong,
*Isometric embedding of Riemannian manifolds in Euclidean spaces*
(p.40):

Theorem 2.3.1. Let $g$ be a smooth complete metric defined in $\mathbb{R}^2$. Then for any compact set $\Omega$ in $\mathbb{R}^2$, $(\Omega,g)$ admits a smooth isometric immersion in $\mathbb{R}^4$.

Since Poznyak's $\mathbb{R}^4$ result is from 1973, and Gromov's $\mathbb{R}^5$ from 1986, it must be that Poznyak's theorem does not improve Gromov's to $\mathbb{R}^4$. Is this because "a compact part of a complete surface" is not the same as a "compact surface"? Can anyone explain this to me? Thanks!

E.G. Poznyak, "Isometric immersions of two-dimensional Riemannian metrics in Euclidean space."

*Russian Math. Surveys*, 28:4 (1973), pp. 47–77.

*Uspekhi Mat. Nauk*, 28:4 (1973), pp. 47–76.

**Answered**. Robert Bryant resolved my confusion:
"A compact surface without boundary can't be identified with a subset of the plane."
It is my understanding now that it remains an open question of whether or not every compact
surface (without boundary) has a smooth immersion in $\mathbb{R}^4$: neither Gromov's nor
Poznyak's results settle this question, and as far as I know there is no known counterexample.

I'm adding the `open-problem`

tag, because although Robert completely answered my specific
question, the question suggested by the title is open.