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 ().
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, :
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 .