Topology and Geometry Topologyand Geometry

Math 6342/7350: Topology and Geometry Sample Preliminary Exam Questions 1. For each of the following topological spaces Xi , determine whether Xi and Xi × Xi are homeomorphic. (a) X1 = [0, 1] (b) X2 = R2 (c) X3 = Z (d) X4 = the middle-third Cantor set 2. Let X = N = {1, 2, 3, . . . }, and equip X with the topology T = {U ⊂ X | (2n − 1) ∈ U ⇒ 2n ∈ U }. That is, U ∈ T if and only if every odd number (2n − 1) that is contained in U has a successor (2n) that is also contained in U . Thus U = {1, 2, 3, 4} ∈ T since the odd elements of U (namely 1 and 3) have successors (2 and 4) that are also contained in U ; on the other hand, V = {1, 2, 3} ∈ / T since the odd number 3 is an element of V but its successor 4 is not. (a) Prove that (X, T ) is locally compact but not compact. (b) Determine (with proof) the connected components of (X, T ), and show that (X, T ) is locally pathconnected. 3. Let Rω denote the set of all infinite sequences of real numbers and let 0 ∈ Rω be the sequence of all zeros. (a) What is the connected component of 0 in the product topology? (b) What is the connected component of 0 in the uniform topology? 4. Prove that if X is Hausdorff, then any compact subset of X is closed. Also give an example of a topological space that is not Hausdorff with a compact subset that is not closed. 5. Let p be an odd prime integer. Define d : Z × Z → R as follows. If m = n, set d(m, n) = 0. If m 6= n, set d(m, n) = 1/(r + 1), where r is the largest nonnegative integer such that pr divides m − n. (a) Prove that d is a metric on Z. (b) With respect to the topology on Z induced by the metric d, is the set of even integers closed? 6. (a) Let X be a path connected topological space and let A be a path connected subset of X. Suppose there exists a continuous map r : X → A such that r(a) = a for every a ∈ A. Prove that r∗ : π1 (X) → π1 (A) is surjective. (b) Let D2 denote the closed unit disk in R2 and notice that the unit circle S1 forms the boundary of D2 . Prove that there does not exist a continuous map r : D2 → S1 such that r(z) = z for every z ∈ S1 . 7. Let Rω denote the set of all infinite sequences of real numbers and let R∞ ⊂ Rω be the set of all sequences that are eventually 0: that is, (x1 , x2 , . . . ) ∈ R∞ if and only if there is N ∈ N such that xn = 0 for all n ≥ N . Determine the closure of R∞ in the product topology, in the box topology, and in the uniform topology. 8. Let L ⊂ R2 be the x-axis and H = {(x1 , x2 ) | x2 > 0} the upper half-plane. Let X = H ∪ L. Given x = (x1 , x2 ) ∈ R2 , let B(x, r) = {y ∈ R2 | d(x, y) < r}, where d is the usual Euclidean metric. Let B1 = {B(x, r) | x = (x1 , x2 ) ∈ H, 0 < r < x2 }. Given x ∈ R and r > 0, let A(x, r) = B((x, r), r) ∪ {(x, 0)}. Let B2 = {A(x, r) | x ∈ R, r > 0}. (a) Show that B = B1 ∪ B2 is a basis for a topology on X. Topology and Geometry (b) Determine (with proof) whether or not the topology T generated by B is first-countable and/or second-countable. (c) Show that (X, T ) is regular. 9. Let M be a smooth manifold and x ∈ M . (a) Define the tangent space Tx M and explain why it is a vector space. (b) Define the tangent bundle T M and explain why it is a smooth manifold. 10. Describe (with justification) the fundamental group of: (a) the 2-sphere S 2 ; (b) the 2-torus T2 ; (c) the real projective plane RP 2 . 11. Let θ and γ be smooth 3-forms on S7 . Prove that Z Z θ ∧ dγ = dθ ∧ γ. S7 S7 Hint: recall that if ω is a smooth k-form and η is a smooth l-form, we have d(ω ∧ η) = dω ∧ η + (−1)k ω ∧ dη. 12. (a) State the Sard theorem. (b) Let f : S1 → S2 be a smooth map. Prove that f cannot be surjective. (c) For a plane P in R3 , let πP : R3 → P denote the orthogonal projection onto P . Suppose that g : S1 → R3 is a smooth embedding. Prove that there exists a plane P for which πP ◦ g is an immersion. 13. (a) Prove that if X is a Hausdorff space, then for every point x ∈ X and every compact subset A ⊂ X, there are disjoint neighbourhoods U 3 x and V ⊃ A. (b) Let X be a locally compact Hausdorff space and give the definition of the one-point compactification X ∗ (you must define both the set and the topology). (c) Use the previous two parts to prove that every locally compact Hausdorff space is regular. 14. Let M be a smooth manifold and S ⊂ M an embedded smooth submanifold. Let p ∈ S and v ∈ Tp M be such that vf = 0 for every f ∈ C ∞ (M ) with f |S ≡ 0. Let ι : S → M be the inclusion map and show that v ∈ dιp (Tp S). Show that this may fail if S is only assumed to be immersed (instead of embedded). 15. Consider the map F : R3 → R2 given by F (x, y, z) = (z 2 − xy, x2 + y 2 ). (a) Find all the critical points of F . (b) Determine all values (a, b) such that F −1 (a, b) is a smooth one-dimensional submanifold of R3 . Q 16. Let [0, 1] have the usual topology. Let R+ denote the nonnegative reals, and define X := α∈R+ [0, 1] with the product topology. Prove that X is not first countable. (Hint: Let A := {(xα ) : aα = 1/2 for all but finitely many α}. Prove that if 0 is the tuple in X with all entries equal to 0, then 0 ∈ A, but no sequence of points in A converges to 0.) 17. Let X be a nonempty compact Hausdorff space. (a) Prove that X is normal. (b) State the Tietze extension theorem. (c) Prove that if X is also connected, then either X consists of a single point or X is uncountable. Topology and Geometry 18. For n ∈ N, let Sn denote the unit sphere in Rn+1 . (a) Prove that Sn is connected and compact for every n ∈ N. (b) Let R∞ be the space of sequences (xi )∞ i=1 of real numbers such that at most finitely many of the xi are nonzero. Embedding Rn into Rn+1 via (x1 , . . . , xn ) 7→ (x1 , . . . , xn , 0), we may view R∞ as the union of the Rn as n ranges over N. Define a topology on R∞ by declaring that a set C ⊂ R∞ is closed if and only if C ∩ Rn is closed in Rn for every n ∈ N. Now let S∞ be the subset of R∞ consisting of the union of the Sn as n ranges over N. Prove that S∞ is connected but not compact in R∞ . 19. Let R` be the real line with the lower limit topology; that is, the topology generated by the basis {[a, b) | a < b ∈ R}. Is R` first countable? Is it second countable? 20. Consider the 2-form ω = z dx ∧ dy + (1 − 2y 2 z 2 ) dy ∧ dz on R3 , where we use the standard (x, y, z)coordinates. (a) Let D = {(s, t) ∈ R2 | s2 + t2 ≤ 1} be the unit disc in R2 , and let f : D → R be given by f (s, t) = (1 − s2 − t2 )s2 , so that f = 0 on ∂D. Let F : D → R3 be given by F (s, t) = (f (s, t), s, t). Then M = F (D) is a smooth submanifold (with boundary) of R3 , and ∂M is the unit circle in the yz-plane. REquip M with the orientation such that F is a smooth orientation-preserving map, and compute M ω. R (b) Let S 2 be the unit sphere in R3 with the usual orientation, and compute S 2 ω. 21. Prove that no two of R, R2 , and R3 are homeomorphic (when equipped with the standard metric topology). 22. Consider the equivalence relation on C2 \ {0} given by putting (z1 , z2 ) ∼ (ωz1 , ωz2 ) for every ω ∈ C \ {0}; write [z1 , z2 ] = {(ωz1 , ωz2 ) | ω ∈ C \ {0}} for the equivalence class of (z1 , z2 ). Recall that the complex projective plane CP 1 is defined as the quotient space of C2 \ {0} by this equivalence relation, so that the elements of CP 1 are the equivalence classes [z1 , z2 ]. (a) Determine (with proof) the fundamental group of CP 1 . (b) Let p be a polynomial in one variable with complex coefficients, and let G : C → CP 1 be given by G(z) = [z, 1]. Show that there is a unique continuous map p̃ : CP 1 → CP 1 such that p̃ ◦ G = G ◦ p; that is, the diagram below commutes. C G p C CP 1 p̃ G CP 1 (c) Show that the map p̃ is smooth when CP 1 is given the standard smooth structure (as a real manifold). 23. Let X and Y be topological spaces, and suppose that f : X → Y is continuous and injective. (a) If X is Hausdorff, is it necessarily true that Y is Hausdorff? If you answer YES, provide a proof. If you answer NO, provide a counterexample. (b) If Y is Hausdorff, is it necessarily true that X is Hausdorff? If you answer YES, provide a proof. If you answer NO, provide a counterexample. 24. Let X and Y be topological spaces. We say that a function f : X → Y is an open map if whenever U is an open subset of X, then f (U ) is an open subset of Y . Prove that if X is compact, Y is Hausdorff and connected, and f : X → Y is a continuous open map, then f is surjective. Topology and Geometry 25. Let M be a smooth manifold and fix p ∈ M . Recall that a tangent vector v ∈ Tp M can be viewed either as a derivation or as an equivalence class of curves. Make each of these precise (define “derivation” and “equivalence class of curves” in this setting), and describe the relationship between the two: given a derivation, state which family of curves it corresponds to, and vice versa. 26. Let G be a Lie group with identity element e. Given v ∈ Te G, show that there is a unique left-invariant vector field X on G such that Xe = v. In addition, prove that X is smooth. 27. Let R` be the real line with the lower limit topology; that is, the topology generated by the basis {[a, b) | a < b ∈ R}. (a) Is R` first countable? Is it second countable? (b) Let L be a line in the plane equipped with the subspace topology it inherits as a subset of R` × R` . Are all of the resulting topological spaces homeomorphic to each other? That is, if L, L0 are two such lines, is L homeomorphic to L0 ? If so, prove it; if not, describe all the possible topologies on L. 28. Let M be a smooth manifold and ω a differential form on M . Prove that if ω has even degree then ω ∧ dω is exact. 3 29. Let G be the  Heisenberg  group; that is, G = R with multiplication given by identifying (x, y, z) with 1 x z the matrix 0 1 y  , so (x, y, z) · (x0 , y 0 , z 0 ) = (x + x0 , y + y 0 , z + z 0 + xy); write 0 for the identity 0 0 1 element with x = y = z = 0. Let X, Y, Z be the left invariant vector fields which evaluate at the identity ∂ ∂ ∂ |0 , Y0 = ∂y |0 , and Z0 = ∂z |0 . Let g = (a, b, c) ∈ G be an arbitrary element of G, and to X0 = ∂x determine Xg , Yg , Zg . 30. The Klein bottle K is the quotient space obtained by starting with the unit square {(x, y) ∈ R2 : 0 ≤ x, y ≤ 1} and then making the identifications (0, y) ∼ (1, 1 − y) for all y ∈ [0, 1] and (x, 0) ∼ (x, 1) for all x ∈ [0, 1]. Use the Seifert-van Kampen theorem to compute the fundamental group of K. 31. Let D2 denote the closed unit disk in R2 . Let v : D2 → R2 \ {0} be a continuous, nonvanishing vector field on D2 . Prove that there exists a point z ∈ S1 at which v(z) points directly inward. Hint: argue by contradiction. 32. Let v ∈ Rn be a nonzero vector. For c ∈ R, define Lc = {(x, y) ∈ Rn × Rm : hx, vi2 = kyk2 + c.} For c 6= 0, show that Lc is an embedded submanifold of Rn × Rm of codimension 1. Here k · k denotes the Euclidean norm on Rm and h·, ·i denotes the Euclidean inner product on Rn . 33. Let (s, t) be coordinates on R2 and let (x, y, z) be coordinates on R3 . Let f : R2 → R3 be defined by f (s, t) = (sin(t), st2 , s3 − 1). (a) Let Xp be the tangent vector in Tp R2 defined by Xp = f∗ Xp . ∂ ∂s |p − ∂ ∂t |p . Compute the push-forward (b) Let ω be the smooth 1-form on R3 defined by ω = dx + x dy + y 2 dz. Compute the pullback f ∗ ω. 34. Let X and Y be topological spaces and let f : X → Y be a map. Prove that f is continuous if and only if for every x ∈ X and every net (zα ) such that (zα ) converges to x, we have that (f (zα )) converges to f (x). Topology and Geometry 35. Recall that a topological space Y is said to be locally compact if for every y ∈ Y , there exists an open neighborhood Uy of y such that Uy is compact. (a) Give the definition of a second countable topological space. (b) Let X be a second countable, locally compact, Hausdorff space. Let X + = X ∪ {∞} be the onepoint compactification of X. Recall that a set V is open in X + if and only if V is open in X or V = X + \ C for some compact set C ⊂ X. Prove that X + is second countable. 36. Let X be a topological space and let A ⊂ X. A retraction r : X → A is a map such that r(x) = x for all x ∈ A. (a) State Stokes’ theorem for smooth orientable manifolds with boundary. (b) Let M be a smooth n-dimensional compact connected orientable manifold with boundary. Prove that there exists no smooth retraction r : M → ∂M . Hint: proceed by contradiction and consider a nonvanishing smooth (n − 1)-form on ∂M . 37. Let S 1 ⊂ C be the unit circle. Let X = R × S 1 and Y = T2 = S 1 × S 1 ⊂ C × C. Define p : X → Y by p(x, z) = (e2πix , z 3 ). Pick a base point x0 ∈ X and let y0 = p(x0 ) ∈ Y . (a) Determine the fundamental groups π1 (X, x0 ) and π1 (Y, y0 ). (b) Determine the subgroup p∗ (π1 (X, x0 )) ⊂ π1 (Y, y0 ). 38. Let g and h be non-abelian two-dimensional Lie algebras. Prove that g and h are isomorphic. 39. Consider the smooth map F : R2 → R3 given by F (θ, φ) = ((2 + cos φ) cos θ, (2 + cos φ) sin θ, sin φ). Let M = F (R2 ) be the 2-torus obtained as the image of F and endowed with the orientation that makes F F ∗ ω and use this to compute R orientation-preserving. Consider the 2-form ω = x dy ∧ dz. Compute 3 ω. Use your answer to determine the volume of the region in R enclosed by M . M 40. Give [0, 1] the usual topology. Let X be a product of uncountably many copies of [0, 1]; view X as the set of tuples (xα ), where α ranges over the nonnegative reals R+ and xα ∈ [0, 1] for all α ∈ R+ . Give X the product topology. Prove that X is not first countable as follows. (a) Let A ⊂ X be the set of tuples (xα ) such that xα = 1/2 for all but finitely many values of α. Let 0 denote the tuple in X with all entries equal to 0. Prove that 0 ∈ A. (b) Prove that no sequence in A converges to 0. 41. (a) State the Urysohn lemma. (b) Let X be a normal topological space. Suppose that X = V ∪ W , where V and W are open in X. Prove that there exist open sets V1 and W1 such that V 1 ⊂ V , W 1 ⊂ W , and X = V1 ∪ W1 . 42. Describe the universal cover of R2 \ {0}, together with the corresponding covering map. (If you prefer, you can work with C \ {0}.) Prove that the covering space you give is the universal cover. 43. Let X be a set with the finite complement topology (i.e. U ⊆ X is open if and only if U is empty or X \ U is finite). Exactly which subsets of X are compact? Give an argument proving that your answer is correct. 44. For each of the following topological spaces Xi , determine whether Xi and Xi × Xi are homeomorphic. Give complete proofs. (a) X1 = R. (b) X2 = R2 . (c) X3 = Z. Topology and Geometry (d) X4 = {0} ∪ { n1 | n ∈ N}. 45. Given n ∈ N and 1 ≤ k ≤ n, recall that Gk Rn is the Grassmannian manifold consisting of the set of k-dimensional subspaces of Rn , endowed with the usual smooth structure. Determine dim(Gk Rn ) and prove that Gk Rn is compact. 46. Let X1 ⊃ X2 ⊃ X3 ⊃ · · · be a nested sequence of nonempty compact connected subsets of Rn . Prove that the intersection ∞ \ X= Xi i=1 is nonempty, compact, and connected. 47. Let A be an annulus bounded by inner circle C1 and outer circle C2 . Define a quotient space Q by starting with A, identifying antipodal points on C2 , and then identifying points on C1 that differ by 2π/3 radians. Use the Seifert-van Kampen theorem to compute the fundamental group π1 (Q). 48. Let G be a Lie group with multiplication m : G × G → G defined by m(g, h) = gh and inversion inv : G → G defined by inv(g) = g −1 . Let e denote the identity element of G. (a) Show that the push-forward map m∗ : Te G ⊕ Te G → Te G is given by m∗ (X, Y ) = X + Y . (b) Show that the push-forward map inv∗ : Te G → Te G is given by inv∗ (X) = −X. (c) Show that m : G × G → G is a submersion. 49. Give R2 the usual topology, and define K := {(x, y) ∈ R2 : x and y are either both rational or both irrational}. Prove that K is a connected subset of R2 . 50. Prove or disprove: A topological space X is Hausdorff if and only if the diagonal ∆ = {(x, x) | x ∈ X} ⊂ X × X is closed. 51. Consider the smooth map F : R2 → R3 given by F (θ, φ) = ((2 + cos φ) cos θ, (2 + cos φ) sin θ, sin φ). Let M = F (R2 ) be the 2-torus obtained as the image of F and endowed with the orientation that makes 2 ∗ RF orientation-preserving. Consider the 2-form ω = x dy ∧ dz. Compute F ω and use this to compute ω. M 52. Determine (with justification) whether or not each of the following smooth maps is an immersion, a submersion, an embedding, and/or a covering map. If it is a covering map, determine the degree of the covering. (a) F : S 1 → R given by F (x, y) = y, where S 1 ⊂ R2 is the unit circle. (b) G : S 2 → RP 2 given by G(x) = [x], where we recall that RP 2 can be defined as the quotient space S 2 / ∼ under the equivalence relation x ∼ −x, and [x] ∈ RP 2 is the equivalence class of x ∈ S 2 . (We think of S 2 as the unit sphere in R3 .) (c) H : R/Z → S 2 given by H([t]) = (cos 2πt, sin 2πt, 0).