Lists of the first fundamental group of spaces.

Question

Here are some list to start with $$\begin{array}{|c|c|c|} \hline \mbox{Space}(S)& \pi_1(S) \\ \hline \mathbb{R}^2&0 \\ \hline \mathbb{S}^1& \mathbb{Z} \\ \hline 1-Torus& \mathbb{Z}\times\mathbb{Z} \\ \hline \end{array}$$ How many more could you add to this list?

Answer 1

Some standard examples:

• Any contractible space ($\mathbb{R}^n, \mathbb{C}^n$, etc.) has $\pi_1 = 1$.
• $\pi_1 S^1 = \mathbb{Z}$.
• $\pi_1 S^n = 1$ for $n > 1$.
• The $n$-torus $T^n = S^1 \times \cdots \times S^1$ has $\pi_1 T^n = \pi_1 S^1 \times \cdots \times \pi_1 S^1 = \mathbb{Z}^n$.
• The closed, orientable surface $M_g$ of genus $g > 0$ has $$\pi_1(M_g) = \langle{a_1, \dots, a_g, b_1, \dots, b_g\;\big|\; [a_1, b_1] \cdots [a_g b_g] = 1\rangle}$$
• The closed, nonorientable surface $N_g$ of genus $g > 0$ has $$\pi_1(N_g) = \langle{c_1, \dots, c_g\;\big|\; c_1^2 \cdots c_g^2 = 1\rangle}$$
• The lens space $L(p, q)$ has $\pi_1 L(p, q) = \mathbb{Z}_p$.
• Real projective space $\mathbb{RP}^n$ with $n > 1$ (for $n = 1$, we have $\mathbb{RP}^1 = S^1$) has $\pi_1(\mathbb{RP}^n) = \mathbb{Z}_2$.
• Complex projective space has $\pi_1 \mathbb{CP}^n = 1$.
• The real Grassmannian manifold $\operatorname{Gr}_k(\mathbb{R^n})$ of real $k$-subspaces of $\mathbb{R}^n$ has $\pi_1 = \mathbb{Z}_2$, aside from the trivial case $n, k = 1$.
• The complex Grassmannian manifold $\operatorname{Gr}_k(\mathbb{C}^n)$ of complex $k$-subspaces of $\mathbb{C}^n$ has $\pi_1 = 1$.
• The Lie group $SO(n)$ for $n > 2$ (for $n = 2$, we have $SO(2) = S^1$) has $\pi_1 SO(n) = \mathbb{Z}_2$.
• The Lie group $SU(n)$ for $n > 1$ has $\pi_1 SU(n) = 1$.
• The complement of a knot $K$ has fundamental group $\pi_1 (S^3\setminus K)$ given by $$\langle{x, y\vert x^p = y^q\rangle}$$ for the $(p, q)$-torus knot. (For example, the trefoil knot is a $(2, 3)$-torus knot, and the unknot is a $(1, 1)$-torus knot.)
• For a finite, connected graph $G$, the fundamental group $\pi_1 G = F_n$ is free, where $n = \#G/T = \#E(G) - \#V(G) + 1$ for a spanning tree $T$ of $G$.

Answer 2

You can find a space with fundamental group $G$ for any group $G$. If $G$ has a group presentation $\left<S | R \right>$, form a bouquet of circles with a fixed basepoint, one circle for each element of $S$, and call this $Y$. Attach $2$-cells to $Y$ along the appropriate loop for each relation $r\in R$ (In particular, if $r=s_1^{\pm 1}\dots s_n^{\pm 1}$, then the attaching map $\partial D^2=S^1\to Y$ will attach $S^1$ to the loop $s_1^{\pm 1}\dots s_n^{\pm 1}\in \pi_1 Y=F(S)$, the free group on S). Then the fundamental group of this space is $G$. This is an application of Van Kampen's Theorem for CW-complexes.

For example, take $G=\mathbb{Z}_n = \left<a | a^n\right>$. The $1$-skeleton is just a circle with a fixed basepoint. Attach a $2$-cell to the $1$-skeleton along the loop $a^n$. Then if this space is $X$, $\pi_1 X = \left<a\right> / \left<a^n \right>=\mathbb{Z}_n$.