Just to compile a list of Fundamental groups, Homology Groups, and Covering Spaces for common spaces like the Circle, n-sphere ( $S^n$ ), torus ( $T$ ), real projective plane ( $\mathbb{R}P^2$ ), and the Klein bottle ( $K$ ).

Fundamental Group

Circle: $\pi_1(S^1)=\mathbb{Z}$

n-Sphere: $\pi_1(S^n)=0$ , for $n>1$

n-Torus: $\pi_1(T^n)=\mathbb{Z}^n$ (Here n-Torus refers to the n-dimensional torus, not the Torus with n holes)

$\pi_1(T^2)=\mathbb{Z}^2$ (usual torus with one hole in 2 dimensions)

Real projective plane: $\pi_1(\mathbb{R}P^2)=\mathbb{Z}_2$

Klein bottle $K$ : $\pi_1(K)=(\mathbb{Z}\amalg\mathbb{Z})/\langle aba^{-1}b\rangle$

Homology Group (Integral)

$H_0(S^1)=H_1(S^1)=\mathbb{Z}$ . Higher homology groups are zero.

$H_k(S^n)=\begin{cases}\mathbb{Z}&k=0,n\\ 0&\text{otherwise} \end{cases}$

$H_k(T)=\begin{cases}\mathbb{Z}\ \ \ &k=0,2\\ \mathbb{Z}\times\mathbb{Z}\ \ \ &k=1\\ 0\ \ \ &\text{otherwise} \end{cases}$

$H_k(\mathbb{R}P^2)=\begin{cases}\mathbb{Z}\ \ \ &k=0\\ \mathbb{Z}_2\ \ \ &k=1\\ 0\ \ \ &\text{otherwise} \end{cases}$

Klein bottle, $K$ : $H_k(K)=\begin{cases}\mathbb{Z}&k=0\\ \mathbb{Z}\oplus(\mathbb{Z}/2\mathbb{Z})&k=1\\ 0&\text{otherwise} \end{cases}$

Covering Spaces

A universal cover of a connected topological space $X$ is a simply connected space $Y$ with a map $f:Y\to X$ that is a covering map. Since there are many covering spaces, we will list the universal cover instead.

$\mathbb{R}$ is the universal cover of the unit circle $S^1$

$S^n$ is its own universal cover for $n>1$ . (General result: If $X$ is simply connected, i.e. has a trivial fundamental group, then it is its own universal cover.)

$\mathbb{R}^2$ is the universal cover of $T$ .