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In Hatcher's book, the torus as a CW complex is constructed so:

enter image description here

But as far as I see, I can follow the gluing instruction also in the following way.

I draw the vertex $p$ and the edges $a$ and $b$ so:

enter image description here

Then I draw this picture on a big sphere, and the big region of the sphere that is out of these small disks will be the 2-cell. As far as I see it is glued as is required for beeing a torus (these small disks will be the two holes of the torus). Is this construction really good? Is it really a torus, or I missed something?

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    $\begingroup$ What is end up with is really something homotopic to $\mathbb S^2$ minus two points and this is not a torus. $\endgroup$
    – user99914
    Oct 8, 2015 at 7:47
  • $\begingroup$ What do you mean by two holes of a torus? $\endgroup$
    – user210387
    Oct 8, 2015 at 7:48
  • $\begingroup$ What do you identify after removing the small open disks $\endgroup$ Oct 8, 2015 at 9:20
  • $\begingroup$ No, the gluing is not correct $\endgroup$ Oct 8, 2015 at 9:36
  • $\begingroup$ @MarianoSuárez-Alvarez Could you show me where is the failure? $\endgroup$
    – mma
    Oct 8, 2015 at 11:56

1 Answer 1

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The construction that you did does not lead to a torus. As mentioned by John in the comments what you get is homotopic to $S^2$ minus two points which is homeomorphic to a cylinder. You can then quotient out the cylinder to get a torus.

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  • $\begingroup$ Are you sure? The cylinder is a 2-manifold, but as far as I see, in my object the point $p$ doestn't have an environment that is homeomorphic to an open 2-disk. $\endgroup$
    – mma
    Oct 8, 2015 at 8:21
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    $\begingroup$ @mma Your space is homotopic to the cylinder as the cylinder is homeomorphic to $\mathbb S^2$ minus two points. $\endgroup$
    – user99914
    Oct 8, 2015 at 8:23
  • $\begingroup$ Oh, yes, I was superficial. I thought of homeomorphism, but you told homotopic equivalence. Yes, my object is really homotopic equivalent to the cylinder. Still I don't see where is the difference between the two kind of glues. $\endgroup$
    – mma
    Oct 8, 2015 at 11:57

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