Examples of fundamental groups

Here is a short list of techniques to find fundamental groups:

1. Find a deformation retraction to a space for which the fundamental group is known
2. The Seifert-Van Kampen theorem
3. Use covering spaces and the path/homotopy lifting theorems (See Munkres for this material)

Here is a short list of examples you can study. Try and prove each using all 3 techniques. In my opinion, the 3rd way is usually the hardest and I can only ever use it to prove simple examples.

1. The circle, the $2$-torus, $S^{1}\times S^{1}\times\cdots\times S^{1}$

2. $\mathbb{R}^{n}$, $\mathbb{R}^{n}$ with one point removed, $\mathbb{R}^{n}$ with $k$ points removed, $\mathbb{R}^{n}$ with a line removed, $\mathbb{R}^{n}$ with several lines removed, $\mathbb{R}^{n}$ with a circle removed, $\mathbb{R}^{n}$ with a circle and line removed, etc.

3. The $n$-sphere, the $n$-sphere with $k$-points removed, the $n$-sphere with a circle removed, the $n$-sphere with a $2$-sphere removed, the $n$-sphere with a $k$-sphere removed, etc.

4. The torus with a point removed, the torus with an arbitrary number of points removed, the solid torus, the orientable genus $2$-surface, the orientable genus $n$-surface, the non-orientable surfaces, the orientable genus $2$-surface with a point removed, etc.

5. The complement of the unknot in $\mathbb{R}^{3}$. The complement of the trefoil in $\mathbb{R}^{3}$. The complement of an arbitrary knot embedded in $\mathbb{R}^{3}$.

6. The $n$-fold dunce cap

7. Real and complex projective space of arbitrary dimensions, an arbitrary Grassmanian $Gr(n,k)$

8. Some of the matrix groups, $GL(n,\mathbb{R})$, $SL(n,\mathbb{R})$, $SO(n,\mathbb{R})$, $Sp(2n,\mathbb{R})$, etc.

This is all I can think of for now, but I feel as though anyone seriously studying algebraic topology should try and work out these examples at some point.

If you happen to know about braid groups or configuration space and are recently learning about the fundamental group, then the following should blow your mind.

The configuration space of $n$ points in a topological space $X$ is usually defined to be,

$$C_{\hat{n}}(X) = \{(z_{1},...,z_{n}) \in X^{n} \; | \; z_{i} \neq z_{j} \; \text{if} \; i\neq j \}$$

A theorem which may be enlightening to your intuitive understanding of $C_{\hat{n}}(X)$ would be the following:

$$C_{\hat{n}}([0,1]) = \coprod_{i=1}^{n!} \Delta_{i}^{n}$$

Where $\coprod$ denotes disjoint union, and $\Delta_{i}^{n}$ denotes the $i^{th}$ copy of the $n$-simplex $\Delta^{n}$. We see that there are $n!$ $\Delta^{n}$'s because the symmetric group (which has order $n!$) acts freely on $C_{\hat{n}}(X)$, permuting the coordinates in each $\vec{z}=(z_{1},...,z_{n}) \in C_{\hat{n}}(X)$.

In fact, we can define the orbit space $C_{n}(X) := C_{\hat{n}}(X) / \Sigma_{n}$ as configuration space modded out by the symmetric group on $n$ elements $\Sigma_{n}$. Then it can be shown that $$\pi_{1}(C_{\hat{n}},\vec{p}) = PB_{n}, \; \text{and} \; \pi_{1}(C_{n},\tau(\vec{p})) = B_{n}$$ Where, $\tau: C_{\hat{n}}(X) \rightarrow C_{n}(X)$ is an $n!$-sheeted covering map called the orbit space projection, and $PB_{n}$ and $B_{n}$ are the pure braid group and braid group on $n$ strands, respectively.

(i) $\pi_1 ( \mathbb R) \cong \{0\}$. This is because homotopy equivalent spaces have isomorphic fundamental groups and the map $h: \mathbb R \times [0,1], (x, t) \mapsto xt$ is a deformation retract mapping all of $\mathbb R$ to the one-point space $0$ and the fundamental group of a one point space is trivial.

(ii) $\pi_1(S^1) \cong \mathbb Z$. You can see this by observing that $\varphi: [0,1] \to S^1, t\mapsto e^{2 \pi it}$ is a generator of the fundamental group that is, $\pi_1(S^1)$ is cyclic with $[\varphi]^n = [\varphi_n]$ where $\varphi_n (t) = e^{2 \pi i nt}$. For a proof see for example Lee's Introduction to Topological Manifolds, page 181.

(iii) The figure eight space: $\pi_1( S^1 \vee S^1) =\mathbb Z \ast \mathbb Z$. To see this you can use van Kampen's theorem to get an isomorphism $\varphi: \ast_i \pi_1(S^1_i) \to \pi_1(\bigvee_i S^1_i)$. See Hatcher's Algebraic Topology, pp 43 for more detail.

Hope this helps.

Two earlier answers omit the study of the fundamental group of an orbit space.

In Topology and Groupoids it is shown in Chapter 11 that for a discontinuous action of a discrete group on a Hausdorff space $X$ (e.g. $G$ is finite) satisfying some good local properties (e.g. $X$ has a universal cover), then the fundamental groupoid of the orbit space $X/G$ is isomorphic to the orbit groupoid of the fundamental groupoid of $X$. So one needs techniques for calculating orbit groupoids, and their vertex groups. These, due to Higgins and Taylor, are given in Chapter 11. As an example, the fundamental group of the symmetric square of a space $X$ is calculated to be the fundamental group of $X$ made abelian.

Notice the key point that if $G$ operates on a space $X$ then it operates on the fundamental groupoid of $X$, but not necessarily on any fundamental group.

It is also useful to generalise the theory of covering spaces to non free actions. Ross Geoghegan in his 1986 review (MR0760769) of two papers by M.A. Armstrong on the fundamental groups of orbit spaces wrote: "These two papers show which parts of elementary covering space theory carry over from the free to the nonfree case. This is the kind of basic material that ought to have been in standard textbooks on fundamental groups for the last fifty years." However Armstrong does not use groupoids, which makes his results seem more technical.