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.