For any $n$ dimensional closed manifold $M^n$, can we find an open covering $\{U\_i\}\_{i\in[2^n]}$ such that $M=\cup U\_i$ and each $U\_i\cong \mathbb R^n$? How about complex manifolds (replacing $\mathbb R^n$ by $\mathbb C^n$)? 



Can any $n$ dimensional (smooth, PL, topological) closed manifold be covered by $2^n$ pieces of $n$ dimensional real spaces?
In the smooth, PL or topological category, you can cover a $n$-dimensional connected manifold by $n+1$ charts. Furthermore, for $k \leq n-3$ a $k$-connected $n$-dimensional manifold can be covered by at most $\lceil (n+1)/(k+1) \rceil$ charts. The reference for both of these statements is Luft's "Covering of manifolds with open cells". The proof of the first statement is elementary, the latter requires engulfing.


