From Harvard math qualification exam, 1990.
Let $X$ be a smooth manifold with an open cover $N<\infty$ sets $\{B_{n}\}^{N}_{1}$ which are contractible. Assume that $$\pi_{0}(B_{n}\cap B_{m})\le k, \forall n,m$$ for all $n$ and $m$. Give an upper bound to the first betti number of $X$.
Recall that the first betti number is defined to be the rank of $H_{1}(X)$. The main trouble is how the conditon on $\pi_{0}$ relates to the bound. My thought (perhaps dumb) is to proceed this inductively. But an arbitrally large genus surface can be covered by a large enough open ball that is connected. So in this case $n=2g, k=0, N=1$.
I asked my professor today, and he gave the hint that this is nothing but Cech cohomology, in which the $B_{i}$s resemble the faces in simplicial homology. He suggest since $X$ has such a cover then it is the same as putting a simplicial structure on $X$. So the first betti number must be bounded. But how large is the bound given $N$ and $k$? Imagine a two hole torus covered by a million small open balls with one next to each other, then $k$ should be 2 if the balls are small enough. But this seems to carry over to a 3-hole torus or a torus of any genus. I suspect there is some fundamental misunderstanding on the statement of the problem or concepts behind, so I venture to ask. A closer look at the Cech cohomology article does not help me to come up with a bound.