# How much of an $n$-dimensional manifold can we embed into $\mathbb{R}^n$?

I observed some naive examples. Spheres, for example, when we cut out one point, can be embedded into $\mathbb{R}^n$. And if we cut out a measure zero set of a projective space, it can be embedded into the Euclidean space of the same dimension. So I wonder if all manifolds can be embedded into a same dimensional Euclidean space when we cut out a measure zero set? Can anyone prove it or disprove it by giving me some counterexamples?

• +1, interesting question! Another way of phrasing it is, how big can we make a chart? I tried to make a title that was easier to parse, but feel free to change it to something else if you want. Mar 6, 2013 at 1:35
• @Zev: In your phrasing, the question is a duplicate of both math.stackexchange.com/questions/11769/… and also of math.stackexchange.com/questions/18083/…. I'm not sure what the policy is on closing because something is provable equivalent to a duplicate ;-). Mar 6, 2013 at 1:46
• One way to get at this question is to put a Riemannian metric on the manifold and then to consider how big the cut locus of a point can be. I believe it is known that the cut-locus of a point has measure zero, although I forget the proof. If this is the case, you could always remove a measure zero set and then use the exponential map at a point to get the required embedding. Mar 6, 2013 at 1:46
• What is cut locus?
– lee
Mar 6, 2013 at 13:16
• Another idea would be to take a Morse function with a unique minimum. Then, take the union of all the ascending manifolds from higher index critical points. This is of codimension 1 and thus of measure 0. I'm pretty sure the complement is the ascending manifold of the minimum and thus diffeomorphic to a ball. Oh, wait, that's the link that Jason gave. Sorry. Mar 20, 2013 at 13:39

This question was essentially answered here, namely, that the cut-locus has measure zero (see the references provided in the link). I assume that your manifold $M$ is smooth, otherwise, I am not sure what notion of measure zero you would be using. I will also assume that $M$ is connected. (If not, apply this argument to each connected component.) Then, put a complete Riemannian metric on $M$, consider the cut-locus $C(p)$ of a point $p\in M$ and the open subset $U(p)=M\setminus C(p)$. The exponential map $\exp_p: T_pM\to M$ restricts to a diffeomorphism $V(p)\to U(p)$, where $V(p)\subset T_pM$ is a certain open subset (diffeomorphic to the open $n$-ball, where $n$ is the dimension of $M$).
One can ask a similar question in the context of connected $n$-dimensional topological manifolds $M$. Instead of removing a measure zero set, one can remove a closed nowhere dense subset. Then in all dimensions but 4 it follows from the results in the book by Kirby and Siebenmann that $M$ contains an open subset $U$ homeomorphic to $R^n$. (Maybe it was also known prior to their work, I am not sure.) I do not know what to say about dimension 4.
One last thing, Sullivan proved that every topological manifold $M$ of dimension $\ne 4$ admits a unique Lipschitz structure, i.e., an atlas where transition maps are locally Lipshitz. This implies that topological $n$-manifolds ($n\ne 4$) have a well-defined notion of measure zero sets (very indirect though). However, I do not know if the residual set in Kirby-Sibenmann will have measure zero in this sense, this goes well beyond my understanding of the work of their work and the one by Sullivan.