Consider the surface K of the cylinder with radius 1 and height 1 and let $B_1 , B_2$ be its bases. Then $K-(B_1 \cup B_2)$ is a 2-dimensional differentiable manifold.
I get that $K-(B_1 \cup B_2) = S^{1}\times(0,1) $ but what are the considered maximal atlas of each manifold?