Turn in 3 problems from Section 1.3 and the extra problems (A-C) below, and read Section 1.3 and 1.A. Note: 1.A is interesting and perhaps good for you to read, but you are not expected to know all of 1.A for this course (but I think you would be able to do some of those problems on an exam)

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, February 18.

Suggested problems:

Section 1.3 # 3, 15, 22, 23, 27.

A. Pick some space that we haven’t found coverings of, and find 3 coverings of it. Prove the 3 coverings are distinct up to isomorphism.

B. Let \(X = S^1 \vee S^1 \). In class Monday the 9th, I gave a three-fold covering that had a loop labeled \(a\) at the lift \(0\) of the basepoint, but the lifts of \(a\) starting at lifts \(1\) and \(2\) of the basepoint were paths and not loops. Look in your notes for that covering space, \(\widetilde{X}\). I chose the maximal tree given by \(bb\) starting at \(0\) and wrote \( p_\ast(\pi_1(\widetilde{X},0)) = \langle a, bab^{-2}, b^2ab^{-1}, b^3\rangle \). Write out what \( p_\ast(\pi_1(\widetilde{X},1)) \) is. According to some change of basepoint theorem, this subgroup of \( \pi_1(X)\) (that you just found) should be conjugate to to the one I gave you. Show explicitly that they are conjugate subgroups. Draw the covering space, too.

C. Do some problem from 1.A.