Take the torus $T=S_1 \times S_1$. Choose two points $x,y \in T$ and define a quotient topology by identifying $x$ and $y$. Let $X$ denote the quotient space. Prove that: a) Compute the fundamental group of $X$. b) Prove that $X$ is not homeomorphic to a surface.
[Math] Quotient space of the torus with two points identified
algebraic-topologygeneral-topology
Best Answer
Here's how I might do it. (In fact, this is just the method suggested in PseudoNeo's comment above, but I wrote this out before I saw that particular comment, so I'm posting it for posterity.)
Consider the torus, and choose your two points $x$ and $y$. The space $X$ in which these two points are identified is homotopy equivalent to the space formed by attaching a copy of $I$ to the torus with endpoints $x$ and $y$, and since the torus is path-connected, this is homotopy equivalent to the wedge sum of a torus and a circle. By van Kampen's Theorem, and the homotopy invariance of $\pi_1$ we now know the fundamental group. Much like Jacob's proof, these homotopy equivalences are very easy to convince yourself of, but perhaps require a little more work to actually prove.
EDIT: My solution to part b) is obviously nonsense, homotopy equivalence doesn't preserve manifold-ness; consider a sphere with an interval attached at the basepoint, for instance. You can show that it's not homeomorphic to a closed compact surface by using van Kampen's theorem, the classification of such surfaces, the fundamental group calculated in part a), and the homeomorphism invariance of $\pi_1$. You can show that it's not even a 2-manifold by looking at Jacob's solution above.