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.