Due Friday, Nov 20

A. Recall the projective plane \(P\) can be described as the union of a Möbius band \(U_1\) and a disc \(U_2\) over their intersection which is the boundary of each. You already computed the inclusion map from the boundary of the Möbius band to the Möbius band. Use this knowledge and Seifert-van Kampen to calculate \(\pi_1(P, basepoint)\) and specify generators and where they come from topologically.

For B and C, let \(U_1 = T^{(1)}_1\) and \(U_2 = D^2\) so that their union is \(T^{(1)}\) and their intersection is \(S^1\).

B. Compute the inclusion induced maps on the fundamental group level from \(S^1\) into \(U_1\) and \(U_2\).

C. Use B to calculate the fundamental group of the torus with SVK.

D. Use a similar tactic to compute the fundamental group of the Klein bottle, using the model for the Klein bottle with one twisted handle and one untwisted handle.

E. Recall, now, that the Klein bottle is secretly \(P\#P = P^{(2)}\). There is a presentation for \(P^{(2)}\)’s fundamental group. Construct an isomorphism between the presentations from D and E.

F. What are your thoughts? Is SVK (and computing the fundamental group) a useful way to distinguish between spaces? Think, are any of the groups you computed isomorphic to \(\mathbb{Z}*\mathbb{Z}\) or \(\mathbb{Z}/2*\mathbb{Z}/2\)?

I will select some problems from those to grade. Please ask me about any questions you have on the problems that are not graded.