Turn in 3 problems from the ones below.

Disclaimer: You should do all of the suggested problems on your own, and I will assume that you do. I would recommend (for the midterm, final, and qualifying exams) that you do many other problems. I am available for help.

Due Wednesday, March 4 Friday, March 6.

Suggested problems:

Hatcher p 131 section 2.1 problems 1 – 10.

A. Recall the \(\Delta\)-complex structure we put on \(T^2\), which was a subdivision of the usual CW-structure on the torus. Prove the topology on the torus coming from the \(\Delta\)-structure is equivalent to the usual topology on the torus. Use the fourth axiom of \(\Delta\)-complexes.

B. In class February 25, I gave a \(\Delta\)-structure on the nonorientable surface \(N_2\). (This is homeomorphic to the Klein bottle). Use it to compute the simplicial homology groups of \(N_2\).

C. On p 102, Hatcher gives a \(\Delta\)-structure on the orientable surface \(M_2\). (It’s the picture with labels \(a,b,c,d\)). Compute the simplicial homology groups of \(M_2\).