# Embedding of a topological manifold

We know the celebrated 'Whitney embedding theorem' for smooth manifold that says any n-dimensional manifold can be smoothly embedded in $\ \mathbb R^{2n} \$. Now my question is: Is there similar version for topological manifolds? that is for which least N(n) any n-dim topological manifold will have a locally flat embedding in $\ \mathbb R^{N(n)} \$. It's clear that for smoothable topological manifolds N will be equal to 2n for $\ \mathbb RP^{2n} \$ can not be embedded(locally flat) in $\ \mathbb R^{2n-1} \$ ie. $\ \mathbb RP^{2n} \$ can not be realized as topological submanifold of $\ \mathbb R^{2n-1} \$ where n is a power of 2.

But for nonsmoothable topological manifolds can this N be made less?

• If anything $N$ would have to be larger, because every smooth manifold is a topological manifold. – Joe Johnson 126 Aug 30 '15 at 11:59
• My questin is particularly for nonsmoothable manifolds. @Joe Johnson 126 – Neel Aug 30 '15 at 12:04
• This might be of interest: mathoverflow.net/questions/34658/… – Joe Johnson 126 Aug 30 '15 at 12:15