Sample Problems for Final Exam (Parts 1 and 2) Exercise 1 ...

22M:201 Fall 06 J. Simon

Sample Problems for Final Exam (Parts 1 and 2) Exercise 1. Questions about the Mayer-Vietoris sequence . . . • Suppose X = A ∪ B, where A, B are open. State the Mayer-Vietoris sequence that relates the various homology groups. • Define the various homomorphisms. • Prove that homomorphism ∆∗ is well-defined. • Prove the M-V sequence is exact. Remark. When you define the homomorphisms and prove exactness, I would like you to talk in the same way as we did in class, and as in the posted handout. It is ok to say things like “α is an n-chain in A and α = β where β is an n-chain in B implies α is an n-chain in A ∩ B” rather than talking in terms of complicated algebra at the chain-group level. In Exercises 2 – 7, use the Mayer-Vietoris sequence to calcuate the homology groups of X. Exercise 2. X = A ∨ B, where the point A ∩ B is a deformation retract of an open neighborhood (to make sure we can apply the M-V sequence to this situation). Exercise 3. X = {(x, y, z) ∈ R3 | x2 + y 2 + z 2 = 1} ∪{(x, y, 0) ∈ R3 | x2 + y 2 ≤ 1}. Exercise 4. X is the union of two M¨obius bands sewn together via a homeomorphism of their boundaries. Exercise 5. X is the union of a 2-sphere and a circle, as shown in Figure 1 below. Exercise 6. X is a closed orientable surface T of genus n, i.e. X = T 2 #T 2 # · · · #T 2 , where there are n summands T 2 . Note. To do the calculation, express X as the union F ∪ D, where D is a disk, F is the complement in X of a small open disk (i.e. F = “punctured X”) and F ∩ D is their shared boundary circle C. You may assume that (a) F deformation retracts to a wedge of n circles, and (b) the generating 1-cycle for H1 (C) is homologically trivial in F . c Simon, all rights reserved

J.

page 1

Figure 1. Union of a 2-sphere and a circle (two views of the same set). Exercise 7. X is the non-orientable surface obtained by replacing D in Exercise 6 with a M¨obius band M. ( So X = F ∪C M, where C = bdy F = bdy M.) Exercise 8. Suppose we are building a cell-complex X. We have, so far, space Y and we adjoin cell D n to Y to finish building X. For simplicity, assume that the attaching map f : bdy D n → Y is an embedding such that we can apply the Mayer-Vietoris sequence to X = Y ∪S D n , where S = bdy D n = Y ∩ D n . Use the Mayer-Vietoris sequence to show how the homology groups of Y and X are related: In particular, show that EITHER βn (X) = βn (Y ) + 1 OR ELSE βn−1 (X) = βn−1 (Y ) − 1. Exercise 9. Under the same hypotheses as Exercise 8, use the long exact sequence for the pair (X, Y ) [instead of using the M-V sequence] to reach the same conclusion: EITHER βn (X) = βn (Y ) + 1 OR ELSE βn−1 (X) = βn−1 (Y ) − 1. Exercise 10. Let D be a subset of S n homeomorphic to a k-cell. ˜ p (S n − D) are all 0. Prove that the homology groups H Exercise 11. Prove that if Σ is a subset of S n homeomorphic to S n−1 , then S n − Σ has exactly two path components. Exercise 12. Suppose Σ ⊂ S n is homeomorphic to S n−1 . Then we know S n − Σ has exactly two path components U, V . Let p be a point in U. We call a point x ∈ Σ arcwise accessible from U if there is a path α : [0, 1] → S n such that almost all of the path is in U, that is α ([0, 1)) ⊂ U, and α(1) = x. PROVE the set of points of Σ that are arcwise accessible from U is dense in Σ. Hint: A small but important part of the result of Exercise 10 says that a [set homeomorphic to] an (n-1)-ball does not separate S n . Use c Simon, all rights reserved

J.

page 2

this to get a path β from p to some point of V , such that the desired path α is part of β. Exercise 13. State the “Excision Theorem” and state the long exact sequence for the a pair (X, A). Include in your statements the definitions of all the homomorphisms. Exercise 14. Compute the homology groups of spheres. In particular, using excision and the long exact sequences for pairs (but not using ˜ k−1 (S n−1 ). ˜ k (S n ) ∼ the M-V sequence), prove that for all n, k, H =H Exercise 15. Exercise 16. Here are several questions involving the idea of degree of a map. • Define the degree of a map f : S n → S n . • Find the degrees of each of the following maps f : S 2 → S 2 , where S 2 is the unit sphere in R3 . You may state the first result without proof, but you should justify all your subsequent assertions. (1) f (x, y, z) = (x, −y, z). (2) f (x, y, z) = (−x, −y, −z). g(x,y,z) (3) f (x, y, z) = kg(x,y,z)k , where g(x, y, z) = (x + .01, y + .02, z + .01). (4) Any map f that has no fixed point. • Prove that every map f : S 2k → S 2k has at least one fixed point or has at least one point that f sends to its antipode. • State and prove the theorem which says, “You can’t comb the hair on a billiard ball”. That is, “There does not exist a function F : S 2 → R3 such that . . . ”.

c Simon, all rights reserved

J.

page 3