I know by [Van Kampen](http://pi.math.cornell.edu/~hatcher/AT/ATchapters.html)'s Theorem that we can obtain $\pi\_1(S\_1 \vee S\_1) = \mathbb{Z} \* \mathbb{Z}$, so I am wondering if we can construct a surface or 3-manifold whose fundamental group is $\mathbb{Z}\_n \* \mathbb{Z}\_2$ or even $\mathbb{Z}\_m \rtimes \mathbb{Z}\_n$.


This might be an interesting problem because I have written semidirect product $\rtimes$ rather than the free product $\*$. A torus knot $K$ is defined in Hatcher as the image of an [embedding](https://en.wikipedia.org/wiki/Embedding) of a map $f : S^1 \to S^1 \times S^1 \to \mathbb{R}^3 \subset S^3$ given by $z \mapsto (z^m, z^n)$ then the fundamental group $\pi\_1(\mathbb{R}^3 - K)$ is $Z\_m \ast Z\_n$ possibly up to some number-theoretic conditions. Hatcher doesn't quite give you the answer.


I think the [semidirect](https://en.wikipedia.org/wiki/Semidirect_product) product $\mathbb{Z}\_m \rtimes \mathbb{Z}\_n$ is unique. We have to specify $\mathbb{Z}\_m \lhd G$ and then $G = \mathbb{Z}\_m \ltimes \mathbb{Z}\_n$.



Find a surface or 3-manifold whose fundamental group is $(\mathbb{Z}/n\mathbb{Z}) \rtimes (\mathbb{Z}/2\mathbb{Z})$
See pp. 449--457 of Peter Scott's article *The geometries of 3-manifolds* for a complete description of all 3-manifolds with finite fundamental group. The article is available on his [website](http://www.math.lsa.umich.edu/~pscott/). There don't seem to be any with dihedral fundamental groups (see Allen Hatcher's comment below), but the fundamental groups of the *prism manifolds* are the *binary dihedral groups*, i.e. non-split central extensions of $D\_{2n}\cong \mathbb{Z}/n\rtimes\mathbb{Z}/2$ by $\mathbb{Z}/2$.


