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 \(M_2\).

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.

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.

Turn in 3 problems from Section 1.3 and the extra problems (A-D) below, and read Section 1.3

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 11.

Suggested problems:

Section 1.3 # 10, 12, 14, 16

A. Show that the quotient map \(S^2 \to \mathbb{R}P^2\) is a covering map. This will help on number 14.

B. Prove this. Let \(p:\widetilde{X}\to X\) be a covering space. The cardinality \(\#p^{-1}(x)\) of the fiber over a point is locally constant. If \(X\) is connected, this is globally constant.

C. Construct a two-sheeted covering of the Klein bottle by the torus.