A subset of the torus s1 times s1 is a punctured torus s1

Question

Honors Examination in Topology 2012

Notation - The following conventions for notation have been followed:

Z = the set (group, ring) of rational integers.

Q = the set (group, field) of rational numbers.

R = the set (group, field) of real numbers.

C = the set (group, field) of complex numbers.

S^n-1 = the unit sphere in Rⁿ.

1. Consider the following collection of subsets of the natural numbers N = {1, 2, 3, . . . ,}: We will say that a proper subset K is closed in N if K = {a_i} and Σ_i 1/a_i is finite. We also take N to be closed. Prove that this collection determines a topology on N and prove that N is connected in this topology. Is it Hausdorff?

2. A function f: (X, d) → (Y, d′) is uniformly continuous if for each ∈ > 0, there is a δ > 0 so that, for all x, y in X, d(x, y) < δ implies d′ (f(x), f(y)) < ∈. Prove that if (X, d) is a compact metric space and f: (X, d) → (Y, d′) is continuous, and then f is uniformly continuous.

3. Consider the equivalence relation on R given by x ∼ y where x ∼ y if x and y are integers, and x ∼ y means x = y if x or y is not an integer. Consider the quotient map p: R → R/∼. Prove that p is a closed mapping. Is R/∼ Hausdorff? Is R/∼ locally compact, that is, for any [x] ∈ R/∼ and open set U with [x] ∈ U, there is a compact set K ⊂ R/∼ with [x] ∈ (interior K) ⊂ K ⊂ U.

4. A continuous closed curve γ: [0, 1] → C - {0}, for which γ(0) = γ(1) = 1 ∈ C has winding number wn(γ) defined by the function γ˜(s) = γ(s)/|γ(s)| = e^ig(s) for a choice of continuous function g: [0, 1] → R and wn(γ) = g(1) - g(0)/2π. Prove that the inclusion S¹ → C - {0} induces an isomorphism of fundamental groups, Z ≅ π₁(S¹, 1) ≅ π₁(C - {0}, 1) and that the homotopy class of the curve [γ] ∈ π₁(C - {0}, 1) determines an integer, which is ±wn(γ).

5. A subset of the torus S¹ × S¹ is a punctured torus S¹ × S¹ - {p}, where p is some point in the torus. Show that the punctured torus has a figure eight (S¹ ∨ S¹) as retract, and consider the homomorphism i_∗: π₁(S¹ × S¹ - {p}, q) → π₁(S¹ × S¹, q) induced by the inclusion. Describe the homomorphism and justify your answer.

6. An embedding of a graph in the plane can be construed as an embedding into the sphere S² and the edges break the sphere up into faces, leading to an occasion to apply Euler's formula. Let K_3,3 denote the complete bipartite graph known as the houses-utilities graph. Prove that K_3,3 is not planar using Euler's formula. From the graph you can get the number of faces, and from the graph you can deduce the minimum number of edges for each face (a face corresponds to a circuit in the graph). Counting edges from the faces leads to a desired contradiction.

7. Let M(S¹) denote the following set, M(S¹) = {{u, v} ⊂ S¹}, that is, the set of all subsets of S¹ of the form {u, v}. Notice that this does not preclude u = v. We can topologize M(S¹) as a quotient of S¹ × S¹. Show that M(S¹) is homeomorphic to the Mobius band, with the boundary of this Mobius band identified with S¹.

8. On Busy Bee World, the entire surface of this spherical planet is covered by a hive that is one cell deep, and the cells take the shape of a hexagon (when all is well) or a pentagon (in a few cases). Prove that there are exactly 12 pentagons under these assumptions.

9. Homology with coefficients can sometimes be useful to distinguish spaces. Compute the homology of RP², the real projective plane, with coefficients in the field Q. Use your calculation to prove that every mapping f: RP² → RP² has a fixed point.

10. A graded vector space is a sequence of vector spaces over a field F, {V₀, V₁, V₂, . . . , V_n, . . .}, and denoted V_•. The Euler characteristic for a graded vector space is defined by χ(V_•) = _i=0Σ^N(-1)ⁱ dim V_i.

Suppose that A_•, B_• and C_• are graded vector spaces for which A_n, B_n and C_n are finite dimensional for all n. Suppose further that there is an integer N > 0 with A_N+k = {0}, B_N+k = {0}, and C_N+k = {0}, for all k > 0. Finally, suppose that there is a sequence of linear transformations which is exact:

0 → A_N →B_N → C_N → A_N-1 → · · ·

→ A₁ → B₁ → C₁ → A₀ → B₀ → C₀ → 0.

(a) Show that χ(B_•) = χ(A_•) + χ(C_•).

(b) The Mayer-Vietoris sequence is an exact sequence associated to a pair of subcomplexes A and B of a finite simplicial complex X, such that X = A ∪ B. Suppose H_k( ) = H_k( ; F) denotes homology with coefficients in the field F. Then the following sequence is exact:

· · · →H_k(A ∩ B) → H_k(A) ⊕ H_k(B) → H_k(X) → H_k-1(A ∩ B) →

          · · · → H₁(X) → H₀(A ∩ B) → H₀(A) ⊕ H₀(B) → H₀(X) → 0.

Use (a) to prove that χ(A) + χ(B) - χ(A ∩ B) = χ(X) = χ(A ∪ B). (This identity resembles a basic property of a measure.)