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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'?

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2 Answers 2

up vote 5 down vote accepted

This is a well-known issue, which one resolves as follows:

First, it is better not to work with some given metric on the manifold, because you don't know precisely how the metric structure (e.g. the balls in the given metric) interact with the locally Euclidean structure. So it is better to first choose a n.h. of a point $p$ which is diffeomorphic to an open disk in $\mathbb R^2$, and then to work with that n.h., which one can now think of very concretely as an open disk in the plane.

For concreteness, suppose it is the open disk of radius $r$ around $0$. Now delete the open disk of radius $r- \epsilon$, for some positive $\epsilon$.

So we are in a situation very similar to you one you described (i.e. $M\setminus B(p,\epsilon)$) except that we know exactly what it is that we deleted (i.e. we are certain that it is a small disk), and we know that there is an open annulus of inner radius $r-\epsilon$ and outer radius $r$ left over.

We can do the same thing at the other point (using Hausdorffness, as you point out).

Now when we glue in the cylinder, think of the cylinder as a circle cross an open interval, say $I$. Now we can choose small open subintervals $I_1$ and $I_2$ at either end of the interval $I$ (so that altogether $I = I_1 \cup \text{ a closed interval } \cup I_2$, with the unions being disjoint). Notice that a circle cross $I_1$ is diffeomorphic to our open annulus from above, and the same for a circle cross $I_2$. (This is just using the fact that the $I_i$ are open intervals, and I leave it as an exercise for you to find the diffeomorphism.)

So now we can glue our cylinder to $M$ minus the two disks by identifying the circle cross $I_1$ with the annulus around one of our deleted points, and the circle cross $I_2$ with the other annulus, using the diffeos of the preceding paragraph.

In summary: we have carefully glued in our cylinder to that there are no sharp corners, by making sure that each end of the cylinder is glued smoothly to an annulus around each of the points that is being deleted.

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I cannot add a comment yet, so I am posting this as a separate answer. Firstly, thank you Matt E for the response. Now, supposing I want to find a diffeomorphism from $S^1\times I_1$ to the annulus, shouldn't the annulus itself be open, i.e., I have to remove a closure of $B(p,r-\epsilon)$. In which case, I can think of the annulus as $S^1 \times (r-\epsilon,r)$ and then the diffeomorphism is just the lift of the usual diffeomorphism between open intervals.

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Dear Dactyl, Assuming I've understood what you're writing correctly, the answer is yes. (In some sense, the content of my diffeomorphism exercise is to give a diffeomorphism between an open annulus in the plane and the product $S^1$ cross open interval.) –  Matt E Dec 17 '10 at 1:39

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