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 \(H_2(\mathbb{R}^3-K)\).

Read Examples 2.36 to 2.43 in the book. We did not have time to cover all of them in class.

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 15.

p 155 # 7, 9abd, 10, 14, 23, 28, 29, 31,

B. Let \(X = S^1 \vee S^1\), the circles labeled \(a\) and \(b\), and form \(Y\) by gluing two 2-cells onto \(X\) by the identifications \(a^4\) and \(a^4b^{-2}a^2b^2a^{-2}\). Compute the homology groups of \(Y\).

C. Recall that \(\mathbb{R}P^3 = e^0\cup e^1 \cup e^2 \cup e^3\). Let \(X\) be the space obtained by attaching a 4-cell to \(\mathbb{R}P^3\) where the composition of the quotient map and attaching map \(\Delta: S^3 \to \mathbb{R}P^3 \to \overline{e^3}/\partial{e^3} \cong S^3\) has degree \(3\). Compute the homology groups of \(X\).

D. Recall that \(\mathbb{R}P^3\) is naturally a subcomplex of \(\mathbb{R}P^4\). Compute the homology of \(\mathbb{R}P^4/ \mathbb{R}P^3\) using cellular homology.

E. Suppose we have a smooth embedding \(f: S^1\times D^2 \hookrightarrow S^3\). Let \(K = f(S^1\times 0)\) and \(N(K) = f(S^1\times D^2)\). We call \(K\) a knot and \(N(K)\) its tubular neighborhood. We call \(S^3 – K\) the knot complement. Compute \(H_n(S^3 – K)\) for all \(n\) using Mayer-Vietoris. Carefully specify generators of all spaces involved.

Read section starting p 128 on equivalence of singular and simplicial homology. Now you know it and may use it.

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 8.

Suggested problems:

Hatcher p 155

# 3, 4, 8, 9c, 12, 27.

On 9c, describe geometrically the generators of the homology groups.

On 8, follow this hint if you wish: first show that \(\widehat{f}: S^2 \to S^2 \) is homotopic to \(z^{\mathrm{deg\ } f}\) where \( f(z):\mathbb{C}\to\mathbb{C} \) is the polynomial you are given. Then there is only one thing to show on this problem. Why?

Read Example 2.31 p 136, and Exercise 1 p 155.

A. Any problem you didn’t do from Homework 9.

Read section starting p 128 on equivalence of singular and simplicial homology. Now you know it and may use it.

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 1.

Suggested problems:

Hatcher p 131 section 2.1 problems 18, 26.

A. Let \(X\) be a topological space and suppose \(f:X\to X\) is a constant map. Show that the induced map on reduced homology \(f_\ast:\widetilde{H}_n(X)\to \widetilde{H}_n(X) \) is the zero map for all \(n\).

B. Let \(x_0\) be a point in a space \(X\). Show that \(H_0(X,x_0) \cong \widetilde{H}_0(X)\). Hint: play with the quotient \( Z_0(X,x_0) / B_0(X,x_0) \), and recall the theorem involving \(\widetilde{H}_0(X)\).

C. Let \(A\) be a subspace of \(X\). Show that \(H_0(X,A) = 0\) iff \(A\) meets each path-component of \(X\).

D. Let \(A\) be a subspace of \(X\). Show that \(H_1(X,A) = 0\) iff \(H_1(A)\to H_1(X)\) is surjective and each path-component of \(X\) contains at most one path-component of \(A\).

E. Find \(\widetilde{H}_n( S^1\times S^1 / \ast \times S^1 ) \) for all \(n\) using the long exact sequence for the homology of a good pair. Here you might need to use the generators of simplicial homology, but you need not specify generators for the homology groups of the quotient. (you may use the fact that simplicial and singular homology are the same)

F. Suppose we have a smooth embedding \(f: S^1\times D^2 \hookrightarrow \mathbb{R}^3\). Let \(K = f(S^1\times 0)\) and \(N(K) = f(S^1\times D^2)\). We call \(K\) a knot and \(N(K)\) its tubular neighborhood. We call \(\mathbb{R}^3 – K\) the knot complement. Compute \(H_n(\mathbb{R}^3 – K)\) for all \(n\). Hint: use homotopy invariance and excision, and argue that \(H_n(\mathbb{R}^3, \mathbb{R}^3 – N(K)) \cong H_n(\mathbb{R}^3, \mathbb{R}^3-K)\).

G. Find \(H_n(S^1 \vee (S^1\times S^1) \vee S^2 \vee S^4)\) for all \(n\) and specify generators. Draw a picture.

Turn in 3 problems from the ones below (problems you have not yet turned in).

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 25.

Suggested problems:

Hatcher p 131 section 2.1 problems 1 – 13.

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.

Turn in 3 problems from Section 1.2 and the extra problems (A-D) below, and read Section 1.2 (it contains some great examples). Think about all the ways you know how to calculate \(\pi_1\) of a space.

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 4.

Suggested problems:

Section 1.2 # 3, 4, 8, 10, 19

A. The (initial) maps \(\varphi_\alpha: G_\alpha \to G\) when defining the free product \(G\) of the groups \(\{G_\alpha\}_{\alpha\in \Lambda}\) are injective.

B. If the free product of groups exists, then it is unique up to isomorphism. (Use the universal property)

C. \(F(\mathcal{A}) \cong F(\mathcal{B})\) if and only if \(\#\mathcal{A} = \#\mathcal{B}\)

D. The Klein bottle \(K\) can be given a CW structure consisting of one 0-cell, two 1-cells (labelled \(a\) and \(b\)), and one 2-cell attached via the loop \(aba^{-1}b\). I claimed in class that \(K\) is homeomorphic to some orientable or nonorientable surface in the list \(\ldots N_2, N_1, M_0, M_1, M_2 \ldots\) Which one is it? Prove it is homeomorphic to \(K\). Note: this will take some creativity, and I’m happy to give a hint.