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, April 22.

p 155 # 21
p 165 # 1, 4

A. Find a CW complex structure for the Klein bottle cross the circle. Compute homology. Compare to Klein bottle’s homology. Think about the example in class of various dimensional tori. Formulate a guess for what is the homology of \(X \times S^1\) in terms of the homology of \(X\).

B. Consider a space \(X\) and the space \(X \times S^1\). Compute the homology groups of \(X\times S^1\) in terms of the homology groups of \(X\) using Mayer-Vietoris.

C. Did you do E last week? If so, consider \(\mathbb{R}^3 – K \subset S^3 – K \). Using a Mayer-Vietoris sequence, find a generator for \(H_2(\mathbb{R}^3-K)\).