Fundamental group quotient of torus

Question

Let $X=(S^1\times S^1)/\sim$, where we identify $(v,w)$ with $(-v,-w)$ (we view $S^{1}$ as a subset of $\mathbb{R}^{2}$). My question is what is the fundamental group of $X$? Is it just $\mathbb{Z}\times\mathbb{Z}$?

Answer 1

I'll write equivalence classes in $X=(S^1\times S^1)/\sim$ with square brackets. Also I'll think of $S^1$ as the unit circle in $\mathbb{C}$.

Define a map $\phi:X\rightarrow S^1\times S^1$ by $$\phi[x,y]=(x^2,xy).$$ Also define $\theta:S^1\times S^1\rightarrow X$ by $$\theta(x,y)=[\sqrt{x},(\sqrt{x})^{-1}\cdot y].$$ You can check that the maps are well-defined, and moreover that they are inverse homeomorphisms.