TITLE: Is it possible to obtain a sphere from a quotient of a torus?
QUESTION [9 upvotes]: 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.

REPLY [8 votes]: Think of the torus $\mathbb{T}$ as the product of two circles: $\mathbb{T} = \mathbb{S}^1 \times \mathbb{S}^1$.
Define the figure-8 subset $E$ of $\mathbb{T}$ by $$E  =  \mathbb{S}^1 \times \{ p \} \cup \{ p \} \times \mathbb{S}^1,$$ where $p$ is any point of $\mathbb{S}^1$.
Then by identifying $E$ to a point, $\mathbb{T}$ becomes homeomorphic to $\mathbb{S}^2$.
It's easy to check that $\mathbb{T} - E$ is an open square $(0,1) \times (0,1)$.  So the quotient space $\mathbb{T}/E$ is a closed square with its boundary identified to a point, hence homeomorphic to $\mathbb{S}^2$.