There is already quite a bit of discussion about this online, but I feel like a lot of this discussion is very confused so please help me understand the following questions:
Does every connected manifold have a finite atlas with connected charts?
There is a book by Werner Greub called "Connections, Curvature and Cohomology - Volume 1" which, on page 20 contains the corollary that every topological manifold has a finite atlas of at most $n+1$ charts, where $n$ is the dimension of the manifold. Of course, these charts could well be disconnected, so I wonder if we can also find a finite atlas consisting of charts whose domains are each connected.
How does this change if we are working with smooth manifolds?
I assume the proof by Werner Greub still works if we impose the additional requirement that charts be compatible in the sense that the transition maps are $C^{\infty}$, leading to the implication that every smooth manifold has a finite atlas consisting of at most $n+1$ charts.
Finally, I came across the dubious paper A. Solecki. "Finite atlases on manifolds", Annales Societatis Mathematicae Polonae. Series I: Commentationes Mathematicae XVII (1974), which I call dubious because it doesn't have a single citation as far as I can tell. Without having fully verified the proof myself, the paper claims that for every smooth connected manifold $M$ of dimension $n$ there exists a finite atlas consisting of at most $2\cdot3^{2n}$ full charts, where a full chart is one that maps onto $\mathbb{R}^n$. Since each chart is a homeomorphism and $\mathbb{R}^n$ is connected, this would mean that the domains of each of these charts is also connected.
Any clarification is much appreciated.
