List of Fundamental Group, Homology Group (integral), and Covering Spaces

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.

S^2 is universal cover of real projective plane RP^2.

\mathbb{R}^2 is universal cover of Klein bottle K.

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