The fundamental group of $(S^1\times S^1)/(S^1\times \{x\})$

Question

What is the fundamental group of $(S^1\times S^1)/((S^1\times\{x\})$ where $x$ is a point in $S^1$? My guess is that $S^1$ is a deformation retract of $(S^1\times S^1)/(S^1\times \{x\})$. Thus $\pi_1(S^1\times S^1/S^1\times \{x\})\cong\pi_1(S^1)\cong\mathbb{Z}$. At the same time, I know that the torus $S^1\times S^1$ does not deformation retract to $S^1$ (since their fundamental groups are different). If the above claim is true, how identifying all points of $S^1\times \{x\}$ with a single point in $(S^1\times S^1)$ makes it possible to deformation retracts $(S^1\times S^1)(/(S^1\times \{x\})$ onto $S^1$?

Answer 1

The space $X = (S^1 \times S^1)/ (S^1 \times \{x\})$ does not deformation retract to $S^1$. It is more useful to think of it as the quotient of $S^2$ where two points are identified, as can be seen from the surface diagram below:

Then for $p,q \in S^2$, we have $S^2 / (p \sim q)$ is homotopy equivalent to a 2-sphere with a line segment glued between $p$ and $q$, which is in turn homotopy equivalent to $S^2 \vee S^1$. Hence, the fundamental group of $X$ is indeed $\mathbb{Z}$.