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.