I understand that a torus is obtained from a sphere by adding a handle. I'm working on a question which is asking if it is possible to obtain a sphere from a quotient of a torus? It seems like this should be possible by perhaps identifying the insides of the torus? But I'm not quite sure how to properly express this.

Help is very much appreciated.

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    $\begingroup$ Does quotient just mean surjective continuous map? Do you know the description of the torus as a square with opposite sides identified? If you identified those opposite sides to a point, I think you'll get a sphere. $\endgroup$ – CJD Jun 10 '14 at 13:17
  • $\begingroup$ That is what I was thinking, but do you actually get a sphere or just a point? $\endgroup$ – Wooster Jun 10 '14 at 13:18
  • $\begingroup$ See this question: math.stackexchange.com/q/809595/4583 $\endgroup$ – Ayman Hourieh Jun 10 '14 at 13:21
  • $\begingroup$ Wooster, you get a sphere, because the interior of the rectangle hasn't changed. This is topologically the same as taking a closed disk and identifying its boundary to a point, which likewise gives you a sphere. $\endgroup$ – Ted Shifrin Jun 10 '14 at 13:36
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    $\begingroup$ Every positive-dimensional compact manifold is homeomorphic to a quotient of any other compact positive-dimensional manifold. $\endgroup$ – user98602 Aug 3 '15 at 18:39

The answer is yes:

Consider the torus sitting in $\mathbb R^3$ like a donut on a table. Then you see that it is invarant by a rotation of $180$ degrees around an horizontal axis. The quotient by such involution is a sphere and the projection is wat is usually called a branched cover (with four branch points).

In general any orientd closed surface covers the sphere via a branched covering.


Think of the torus T as the product of two circles: T = S1 x S1.

Define the figure-8 subset E of T by E = S1 x {p} union {p} x S1, where p is any point of S1.

Then by identifying E to a point, T becomes homeomorphic to S2.

It's easy to check that T - E is an open square (0,1) x (0,1). So the quotient space T/E is a closed square with its boundary identified to a point, hence homeomorphic to S2.


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