Due Friday, Nov 20

A. Recall the projective plane \(P\) can be described as the union of a Möbius band \(U_1\) and a disc \(U_2\) over their intersection which is the boundary of each. You already computed the inclusion map from the boundary of the Möbius band to the Möbius band. Use this knowledge and Seifert-van Kampen to calculate \(\pi_1(P, basepoint)\) and specify generators and where they come from topologically.

For B and C, let \(U_1 = T^{(1)}_1\) and \(U_2 = D^2\) so that their union is \(T^{(1)}\) and their intersection is \(S^1\).

B. Compute the inclusion induced maps on the fundamental group level from \(S^1\) into \(U_1\) and \(U_2\).

C. Use B to calculate the fundamental group of the torus with SVK.

D. Use a similar tactic to compute the fundamental group of the Klein bottle, using the model for the Klein bottle with one twisted handle and one untwisted handle.

E. Recall, now, that the Klein bottle is secretly \(P\#P = P^{(2)}\). There is a presentation for \(P^{(2)}\)’s fundamental group. Construct an isomorphism between the presentations from D and E.

F. What are your thoughts? Is SVK (and computing the fundamental group) a useful way to distinguish between spaces? Think, are any of the groups you computed isomorphic to \(\mathbb{Z}*\mathbb{Z}\) or \(\mathbb{Z}/2*\mathbb{Z}/2\)?

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, Nov 13

A. Find an example of a retraction that does not come from a deformation retraction. Hint: fundamental group.

B. Let \(M\) denote the Möbius band and \(A\) denote the annulus. Pick basepoints and calculate their fundamental groups. You should get isomorphic groups.

C. For \(X\) the Möbius band or annulus, pick one component of the boundary, and pick a basepoint. Calculate the induced homomorphism on the fundamental group (from boundary into \(X\)). Why does this calculation imply that the annulus and Möbius band are not homeomorphic?

D. Suppose \(f: X \to Y\) is a homotopy equivalence with homotopy inverse \(g: Y \to X\). It may be the case that a chosen basepoint \(x\in X\) satisfies \(g(f(x)) \neq x\). Find a group isomorphism \(\phi: \pi_1(X,x) \to \pi_1(X, g(f(x))\) that you can use to make this equation work: \( g_\ast \circ f_\ast = \phi\). Conclude that homotopy equivalences induce isomorphisms on the fundamental group, even if they change basepoints.

E. Classify the letters A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z up to homotopy type. Little justification is needed (because of problem F, below).

F. Let B denote the topological space given by the letter B. B with the subspace topology from \(\mathbb{R}^2\) is a compact metric space. Let the basepoint be one of the points where the two loops of B overlap. Let n and s be points on the upper half of B and bottom half of B, respectively. So B-n and B-s are open. Use the Lebesgue number lemma to show any loop in B decomposes (up to homotopy) as a finite concatenation \(f_1 * \cdots * f_k\) where each \(f_i\) is a loop in either top loop or bottom loop of B. Suppose B deformation retracts onto B-s (which deformation retracts to a circle). To the best of your ability, show that there are nontrivial loops in B that die when you map them into the fundamental group of B-s. Try to justify that B is not homotopy equivalent to the letter A.

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 5

3.5.8, 3.5.9, 3.5.11, 3.8.1, 3.8.3

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 30

3.2.3, 3.3.5 through 3.3.8, From Section 3.11: #11, 12, 22

A. If \(f:(X,x) \to (Y,y)\) is a continuous map of pointed topological spaces, verify that \(f_\ast: \pi_1(X,x) \to \pi_1(Y,y)\) is a group homomorphism. I showed you the steps in class. Justify them.

B. If \(f:(X,x) \to (Y,y)\) and \(g:(Y,y) \to (Z,z)\), verify that \(g_\ast \circ f_\ast = (g\circ f)_\ast\).

C. If \(f:(X,x) \to (X,x)\) is the identity map, prove \(f_\ast\) is the identity map on the fundamental group.

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 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.