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, February 25.

Suggested problems:

Hatcher 1.3.17 and 18

A. On p 58, pick one of the covers from (7) – (14) and find \(\mathrm{Aut}(p)\). Is it a normal cover?

B. Let \(p:\widetilde{X}\to X\) be a covering with \( \widetilde{x}, \widetilde{x}’ \in p^{-1}(x)\). Let \(G = \pi_1(X,x)\) and \(H = p_\ast(\pi_1(\widetilde{X},\widetilde{x}))\). Let \(N(H) \) denote the normalizer of \(H\), i.e. the largest subgroup of \(G\) in which \(H\) is normal. Let \(\widetilde{\gamma}\) be a path in \(\widetilde{X}\) from \(\widetilde{x}\) to \(\widetilde{x}’\). Prove if \(p_\ast([\widetilde{\gamma}]) \in N(H)\), then there exists an \(f\in \mathrm{Aut}(p)\) such that \(f(\widetilde{x}) = \widetilde{x}’\). Is the converse true?

C. Let \(f:\mathbb{Z}^4 \to \mathbb{Z}^3\) be a linear transformation. Let \(\{a,b,c,d\}\) and \(\{x,y,z\}\) be bases for the domain and codomain respectively. Suppose that the matrix representing \(f\) with respect to these basis is \( \begin{pmatrix} 7 & -14 & 24 & 41\\ 2 & -4 & 4 & 6 \\ 4 & -8 & 10 & 16 \end{pmatrix} \)

Find the groups \( \mathrm{im}(f), \mathrm{ker}(f), \mathbb{Z}^4/\mathrm{ker}(f), \mathrm{\ and\ } \mathrm{coker}(f) \) and give generators for them in terms of \(a,b,c,d,x,y,z\).

D. Any problem from Homework 4 or 5 that you or I didn’t already do.