Embedding 3-manifolds in Euclidean space

Question

By the Whitney Embedding theorem, every 3-manifold can be embedded in $\mathbb{R}^6$. It's my understand that it's an interesting problem to see which 3-manifolds embed in $\mathbb{R}^4$; some do and some don't. My question is: are there known examples of 3-manifolds which don't even embed in $\mathbb{R}^5$ -- where it's necessary to go to $\mathbb{R}^6$? Or does every one embed in $\mathbb{R}^5$? Any references for this sort of stuff?

Answer 1

You can find some discussion and references on page 136 of the book by Lee "Introduction to smooth manifolds". In particular one of the result quoted is that Wall showed in 1965 that every 3-manifold can be embedded in$\mathbb{R}^5$. It also says that while the optimal immersion dimension is known for each $n$ and given by $2n-a(n)$, where $a(n)$ is the numer of 1's in the binary expression for $n$, the best possible embedding dimension is only known for some values of $n$.

These are the works mentioned in the book:

Wall, C.T.C.: All 3-manifolds imbed in 5-space. Bull. Am. Math. Soc. (N.S.) 71, 564–567 (1965)

Osborn, Howard: Vector Bundles, vol. 1. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York (1982)