March 23rd, 2011, 06:57 AM  #1 
Newbie Joined: Nov 2010 Posts: 28 Thanks: 0  Algebraic Topology Help
Any help with these would be great. 1. (a) Describe all possible covering maps X > RP^2 x RP^2 (b) Describe all possible covering maps X > RP^2 2. Show that there is a covering map from the cylinder to the Mobius band 3. Find a two sheeted covering map 
March 24th, 2011, 09:39 PM  #2 
Senior Member Joined: Jun 2010 Posts: 618 Thanks: 0  Re: Algebraic Topology Help
Turloughmack, For the first problem, it helps to know the fundamental group of the real projective plane. Once you recall this, you will want to look for all possible subgroups to find the distinct covering spaces. For the second and third problems, visualize the Möbius strip as a square with vertical sides identified in opposite directions. Notice that if you "cut" out a strip from the middle of this square, you will end up with two pieces homeomorphic to open subsets of the cylinder, after making identifications. Now, write down "geometrically" what is going on. You will have to check that the map is indeed a covering map. You can construct an nsheeted covering in this way for any even n, using only cylinders. Ormkärr 
March 25th, 2011, 07:52 AM  #3 
Newbie Joined: Nov 2010 Posts: 28 Thanks: 0  Re: Algebraic Topology Help
For Q.1 (a) I know that the fundamental group of the projective plane is Z/2 and that it contains two subgroups. S^2 > RP^2 is the universal cover. What are the other covering maps? (B) The fundamental group is Z/2 x Z/2 which is in fact the Klein 4 group. I know that the whole group is its own covering map and the trivial group is the universal cover but again what are the other covering maps? 
March 25th, 2011, 09:33 AM  #4 
Senior Member Joined: Jun 2010 Posts: 618 Thanks: 0  Re: Algebraic Topology Help
Turloughmack, Just take all possible subgroups. These will be covering maps. For a), I think you have them nailed down. For b), there should be five subgroups, including the trivial ones. For instance, {(0,0), (0,1)} and {(0,0), (1,1)}, etc. Ormkärr 

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