MT A532 General Topology II - Final Exam

1. Show that homotopy equivalent spaces have the same fundamental group, and show that their respective sets of path components are in one-one correspondence. If you find this second part confusing, just prove the special case that any space homotopy equivalent to a path connected space is path connected, for partial credit.
2. Show there is no retraction from the Möbius band to its boundary circle.
3. Let X be the Klein bottle and let G denote its fundamental group.
   1. Realize X as a quotient space of [0,1] x [0,1] by identifying certain points of the boundary square.
   2. Compute a presentation for G using the realization from the previous problem. Explain how you know your answer is correct.
   3. By taking two adjacent copies of the square from (a), construct a covering of X by the torus. What conclusions can you draw about G from this? The more you say, the better.
   4. The Klein bottle is also the connected sum (p.257) of two copies of RP^2. Use this to compute a different presentation of G. For extra credit, give an isomorphism between the two presentations.
   5. Is X a K(G,1)? Why or why not?
   6. Are there any finite order elements in G? If so, give at least one. If not, prove why not.
4. Use covering space theory to exhibit explicit free subgroups of every finite rank inside the free group on two generators. Explain carefully why your answer is correct. For each rank n for which you are able to construct such a subgroup, you will get about 15/n^2 points of credit.