From Hatcher Corollary A.10. the (global) immersion for an $n-$dimensional CW complex is possible in some $\mathbb{R}^N$. I have started with $M(G,n)$ (Moore space of type $(G,n)$, $G$ is cyclic finite group, $n\geq 2$) and have checked that by this Corollary, $M(G,n)$ can be (globally) immersed in $\mathbb{R}^{2n+1}$. My question is as follows:
Given a finite CW complex of dimension $n$, can it be locally immersed into $\mathbb{R}^{2n}$? By local immersion I mean in the sense of Spivak, i.e., every point has a neighborhood where the function is one-one.
Thanks in advance!!