I am trying to find the universal cover for the following spaces:
The closed unit disk with the identification p mapping points on the boundary (cos(θ), sin(θ)) to (cos(θ+2π/n), sin(θ+2π/n)).
(a distinct space for each n)
I know that the universal cover cannot be the closed unit disk itself along with the given identification, since p-1((0,0)) = (0,0) while p-1(a boundary equivalence class) = two points, and for path-connected space (which the closed unit disk is), p-1 must be of the same degree.
My thought was to make the covering space a "spikey disk", where you would take the interior of the disk along with one point on the boundary for each equivalence classes (for example with n=2 the equivalence class [(1,0)] would consist of {(1,0),(-1,0)} so you would take add (1,0) to the interior of the disk and leave out (-1,0)).
The map would then be the interior to the interior and the "spikey" points on the boundary to the boundary equivalence classes. To me this looks like a homeomorphism between the quotient space and the spikey ball and therefore would be a on-sheet covering.
However, the fundamental group of my "spikey" ball is trivial since it is star convex, but the fundamental group of each space is not, I believe (except for n=1) and homeomorphic spaces have the same fundamental groups.
Therefore, if I could I get some help finding the universal cover and an explanation of why the spikey ball is not homeomorphic to the quotient space, it would be greatly appreciated.
Also if someone could flesh out the explanation of Fundamental Group on Quotient of Unit Disk, it would again be greatly appreciated.
