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I have a doubt on the construction of the $n$-holed torus as seen on Spivak's Differential Geometry book. Spivak gives a very good argument on how to construct it: take the usual torus $\mathbb{T}^2=S^1 \times S^1$ and cut out a hole on it throwing away all the points on a certain circle. Then he says to take those things, take their disjoint union and identify the points on the boundaries of these holes.

Well, the argument is pretty straightforward and if I understood well it will amount just to take one disjoint union and then using the quotient by some relation $\sim$ to glue all the parts. I'm just not understanding how to describe it.

For instance, the "torus with hole" which Spivak calls a handle seems to be just $\mathbb{T}^2 - C$ where $C\subset \mathbb{T}^2$ is the subset we want to remove from the torus. However, my first problem is here. What exactly is that subset that we want to remove? Spivak says "a circle", however, is it any circle? I've got a little confused on how we describe that explicitily.

After that, Spivak says that we should take $n$ copies of this and take their disjoint union. That's fine, no problem here, however then he says to identifiy the points on the "boundaries". I know we would have to make an equivalence between points to glue the copies of the torus, but again how to make this explicit?

Is this idea of "a handle" more general? I mean, given $n$ topological manifolds $M_1, \cdots M_n$ can we always make "holes" on each of then, take the disjoint union and finally glue them together identifiying the points on the holes?

Sorry if this question is too basic inside the subject of differential geometry, but I'm getting started now with it, so I'm still "getting used to the ground".

Thanks in advance for your help.

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Google search the term "connected sum" –  Chris Gerig Apr 20 '13 at 15:14
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The book "Differential Manifolds" by Antoni Kosinski has a solid treatment of connect-sum. –  Tim kinsella Apr 22 '13 at 1:06
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Well, the argument is pretty straightforward and if I understood well it will amount just to take one disjoint union and then using the quotient by some relation ∼ to glue all the parts. I'm just not understanding how to describe it.

Here's how I think of this construction (using donuts!). Intuitively, we know a donut is like a torus. Well, a torus is really just the surface of the donut. So let's say we have a powdered donut, so that the 'torus' is the 'powder' on the donut's surface. If we want to get a two-holed torus, we can almost just stick two powered donuts together. But we have a problem: the surfaces aren't actually connected - they're just touching. So we can take a small bite out of each donut, and stick the bit parts together.

We are creating something like (but one donut fewer) the picture I ripped from google-images below, and which is from a wikipedia article I'll link you to in a moment:

enter image description here

This is what we mean by 'identifying the two circles - we stick the two donuts together so that the powdered surfaces touch and connect.

This is an example of a 'connected sum.' You can read more about that here.

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