I am trying to work through Boothby's An Introduction to Differentiable Manifolds on my own and, embarassingly, have got stuck at the very first chapter. At the end of section 4, chapter 1 (called: Further examples of manifolds: cutting and pasting), there's this question:
Prove that adding a handle to a 2-manifolds in the fashion described above for $S^2$ and $T^2$ actually does give a 2 manifold
Unfortunately, Boothby gives no formal definitions for cutting and pasting so I have no idea about how to show that a sphere with a handle will be locally euclidean. For example, what if the handle is attached to the surface such that at the joint (a circle) the two surfaces meet sharply, so that for points on the joint we cannot find a normal.
So, my questions are:
Where can I find the operations of cutting formalised (I realise that pasting is connected to quotient topology, so I am studying that now). And, how does one answer this question rigorously?
Edit Upon further thought, 'cutting' could be this: since a manifold $M$ is metrizable, we can remove an open disk (open so that I get a boundary in the resulting manifold) around a point, i.e., we consider $M - B(p,\epsilon)$. So, given two dissimilar points, we remove two non-intersecting open disks (we can do this because a manifold is hausdorff). Now, how do I 'paste' the ends of the cylinder 'smoothly'?