fundamental group of torus minus a point

Question

I want to calculate fundamental group of torus minus a point using Van-Kampen Theorem.I know that result is $\Bbb Z* \Bbb Z$.

I proved to consider like open $U$ the torus minus a disk $S$ that contain the point and like open $V$ the internal part of torus, then $U$ intersected $V$ is the internal part of torus minus disk $S$. Is this right? And then what are fundamental groups of $U$, $V$ and $U$ interseced $V$?

Answer 1

Think of this space as the closed unit square with sides properly identified, and the $(1/2,1/2)$ point removed.

Take $U$ to be an open ball around $(1/2,1/2)$. Consider a ball $B$ around $(1/2,1/2)$ that sits within $U$, and let $V$ be the complement of $B$.

So, this means that:

• $U$ is an open annulus
• $V$ is homotopic to a wedge of two circles
• $U\cap V$ is also an open annulus.

Because the loops in $U\cap V$ are the same loops in $U$, this means that the fundamental group is

$$\underbrace{\mathbb{Z}}_{U}\ast_{\mathbb{Z}}\underbrace{(\mathbb{Z}\ast\mathbb{Z})}_{V} \cong \mathbb{Z}\ast\mathbb{Z}.$$

You should fill in all the details, here!

• Why is $V$ homotopic to a wedge of two circles?
• Why does $V$ have fundamental group $\mathbb{Z}\ast\mathbb{Z}$?
• Why does the “$U$” part of the free product die in the amalgamation?