Topological manifolds which cannot be embedded into any Euclidean space

Question

I take the definition that a (topological) manifold is a topological space which is second-countable, Hausdorff, and locally Euclidean. Munkres proves that every compact manifold can be embedded into some Euclidean space.

What is an example of an (topological) $m$-manifold which cannot be embedded into $\mathbb{R}^n$ for any $n\in\mathbb{N}$? What is the smallest $m\in\mathbb{N}$ for which such an $m$-manifold exists?

Answer 1

There is no such manifold, and in fact invoking dimension theory shows that more is true. In the same book (Munkres' Topology), exercise $\S$50.7 (p. 315 in my edition) is to prove the following:

Every [topological] $m$-manifold can be imbedded in $\Bbb R^{2 m + 1}$.

This result is a corollary of the following theorem, whose proof is in turn the content of the somewhat technical multi-part problem $\S$50.6:

Theorem Let $X$ be a locally compact Hausdorff space with a countable basis, such that every compact subspace of $X$ has topological dimension at most $m$. Then $X$ is homeomorphic to a closed subspace of $\Bbb R^{2m+1}$.

I do not know whether the dimension $2 m + 1$ can be sharpened to $2 m$.