If I start with a, say, 3-CW complex $X$ which can be embedded in $\mathbb{R}^5$, I can get a neighbourhood $U$ of $X$ which has the same homotopy type of $X$. Then $U$ is a $5-$ dimensional open manifold. Can I get a close manifold (compact without a boundary) $M$, of dimension $6$ (or some higher dimension) such that $M$ and $X$ have the same homotopy type?



How can I construct a closed manifold from a finite CW complex?
Take $X=S^3$. Then no closed manifold of dimension at least 6 has the same homotopy type.


