Due Friday, October 9

Exercises 2.9.25 to 2.9.33.

Be sure and give all the details. For example, I might say, “You can check that…” But on your homework, you can’t say that. You have to check that…

The author’s hint on #26 might be confusing. Here’s a simpler version: think polar coordinates. Any point in \(D^2\) is \(r e^{i\theta}\) where \(r\) between 0 and 1. The “coning” refers to the fact that all of the angles \(\theta\) determine the same point if \(r=0\). This is a certain type of quotient space called the cone: for a space \(X\), the cone is \(CX := X\times[0,1] / (x\times 0 \sim y\times 0:\ x,y\in X)\). The disc is the cone of a circle, up to homeomorphism. The author is asking (more or less), for you to extend an isotopy of the homeomorphism of the circle to an isotopy over the disc by using the \(r\) parameter to shrink inwards.

I will select some problems from those to grade. Please ask me about any questions you have on the problems that are not graded.

Due Friday, October 2, in class

Read exercises 2.9.1 – 2.9.8 (it walks you through the classification of 1-manifolds)

Do exercises 2.1.1, 2.1.5, 2.1.10, 2.2.5, 2.9.18, 2.9.20, 2.9.22, and A, B, and C below

I will select some problems from those to grade. Please ask me about any questions you have on the problems that are not graded.

A. Let \(Q = \mathbb{R}^3 – \{0\} / \sim\) and \(P = S^2/ \sim’\) where nonzero \(x,y \in \mathbb{R}^3\) are equivalent (\(x\sim y\)) if there is a nonzero \(\lambda \in \mathbb{R}\) such that \(x = \lambda\,y\), and \(w, z\in S^2\) are equivalent (\(w\sim’ z\)) if \(w = -z\). (Give \(Q\) and \(P\) the quotient topologies). Construct a homeomorphism \(Q\approx P\). We call \(Q\approx P\) the real projective plane and denote it \(\mathbb{R}P(2)\).

B. Recall the handle decomposition of the Möbius band \(M\) from class: it had one 0-handle and one 1-handle with a half twist. The boundary of \(M\) is a circle; let \(f:S^1 \to \partial M\) be a homeomorphism. Attach a 2-handle \(h^2 = D^2 \times D^0\) to \(M\) by a homeomorphism \(g: \partial D^2 \times D^0 \to \partial M\). Here I am thinking of \(\partial D^2\times D^0\) as identical to \(S^1\) and so \(g\) is really \(f\). Show that the handlebody \(M\cup h^2\) is homeomorphic to \(\mathbb{R}P(2)\).

C. Prove \(\mathbb{R}P(2)\) is a connected, compact surface without boundary.

Hint for B:

Let \(D^2\) have the equivalence relation \(x\sim y\) if \(x=-y\) and both \(x,y\) are on the boundary of \(D^2\).

Prove \(D^2/\sim \approx P\) from A.

Think of an annular neighborhood of the boundary of \(D^2\). It is a subset of \(D^2\) whose complement is a disc. If you can show the neighborhood of the boundary maps to a Möbius band in \(D^2/\sim\), you’re done! Why? So show it’s a Möbius band and be done.

Due at the beginning of class, September 25.

Read Chapter 1.8. Do Exercises 1.9.46, 1.9.48, 1.9.49, 1.9.53, 1.9.58, 1.9.61, 1.9.63a, 1.9.75, 1.9.76, 1.9.77

I will select some problems from those to grade. Please ask me about any questions you have on the problems that are not graded.

Due at the beginning of class, September 18.

Exercises 1.4.2, 1.5.4, 1.5.5, 1.9.26, 1.9.31, 1.9.38, 1.9.39, 1.9.44, 1.9.68

I will select some problems from those to grade. Please ask me about any questions you have on the problems that are not graded.

Due Friday, September 11, in class

Exercises 1.1.2, 1.2.4, 1.2.5, 1.2.8, 1.2.13, 1.3.4, 1.9.4, 1.9.8, 1.9.11, 1.9.12

I will select some problems from those to grade. Please ask me about any questions you have on the problems that are not graded.