Find a space whose fundamental group is $\mathbb Z/2 × \mathbb Z$

Question

Find a space whose fundamental group is

i) $\mathbb Z/2 × \mathbb Z$

ii) $\mathbb Z/2 ∗ \mathbb Z$

Here, $\mathbb Z$ is the set for integers. And $*$ is the free product defined as $F(G \amalg H)/\cong,$ $\times$ is the Cartesian product where $A \times B$ has pairs $(a,b)$ such that $a\in A, b \in B$

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My thoughts:

The first one could be $\pi(X \times S^1$) where $X$ has a fundamental group of $\mathbb Z/2$, but I don't think I know how to derive a space with such fundamental group. Can you give me an example of such space, and how to derive such space like that (with the $/2$)?

Thank you in advance!

Answer 1

This is somewhat too long for a comment, but not an answer yet:

Here is a starting hint: The $2$-dimensional real projective space $$\mathbb{R}P^2:=S^2/(-x \sim x)$$ has fundamental group $$\pi_1(\mathbb{R}P^2) \cong \mathbb{Z}_2.$$ To derive this you could look at a unit square $[0,1]^2 \subseteq \mathbb{R}^2$, use the right identifications on the boundary and apply Van Kampen.

Alternatively, you could use covering space theory and a short exact sequence argument. I'll elaborate the answer if needed.