# fundamental group of torus minus a point

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 $$V$$?

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?