Just to compile a list of Fundamental groups, Homology Groups, and Covering Spaces for common spaces like the Circle, n-sphere (), torus (), real projective plane (), and the Klein bottle ().

# Fundamental Group

Circle:

n-Sphere: , for

n-Torus: (Here n-Torus refers to the n-dimensional torus, not the Torus with n holes)

(usual torus with one hole in 2 dimensions)

Real projective plane:

Klein bottle :

# Homology Group (Integral)

. Higher homology groups are zero.

Klein bottle, :

# Covering Spaces

A universal cover of a connected topological space is a simply connected space with a map that is a covering map. Since there are many covering spaces, we will list the universal cover instead.

is the universal cover of the unit circle

is its own universal cover for . (General result: If is simply connected, i.e. has a trivial fundamental group, then it is its own universal cover.)

is the universal cover of .

is universal cover of real projective plane .

is universal cover of Klein bottle .