My Math Forum  

Go Back   My Math Forum > College Math Forum > Topology

Topology Topology Math Forum


Reply
 
LinkBack Thread Tools Display Modes
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
Turloughmack is offline  
 
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 n-sheeted covering in this way for any even n, using only cylinders.

-Ormkärr-
 is offline  
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?
Turloughmack is offline  
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-
 is offline  
Reply

  My Math Forum > College Math Forum > Topology

Tags
algebraic, topology



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Algebraic Topology Turloughmack Topology 1 March 29th, 2011 12:48 PM
Algebraic Topology Turloughmack Topology 1 March 3rd, 2011 06:56 PM
Algebraic Topology Turloughmack Topology 0 February 1st, 2011 09:28 AM
algebraic topology parnaz Topology 1 December 20th, 2006 11:28 AM
algebraic topology parnaz Topology 3 December 4th, 2006 01:28 PM





Copyright © 2018 My Math Forum. All rights reserved.