Middle East Technical University Spring 1998 Math 422 Elementary Topology and Geometry

MIDTERM II

MAY 12, TIME: 90 MIN

SOLUTIONS

Problem 1. (7 pts) Prove that if f: A → ℝ^{m} is a continuous map and A ⊂ ℝ^{n} is path connected, then so is f(A).

Solution: We need to prove existence of a path connecting every pair of points x_{0}, x_{1} ∈ f(A). Let x_{i} = f(x′_{i}), i = 0, 1, then path connectedness of A implies that there exists a path, γ: [0, 1] → A, connecting x′_{0} and x′_{1}. The composition f ∘ γ: [0, 1] → f(A) is a path connecting x_{0} and x_{1}.

Problem 2. (5 pts) Determine the connected components of A = (ℚ × ℝ) ∪ (ℝ × {0}) ⊂ ℝ^{2}.

Solution: A is path connected, thus is connected, so there is only one connected component, A itself. A path connecting any pair of points, x, y ∈ A, is obvious. It consists of tree line segments, two of them connect x = (x_{1}, x_{2}) with x′ = (x_{1}, 0) and y = (y_{1}, y_{2}) with y′ = (y_{1}, 0), and the third segment connects x′ with y′.

Problem 3. (10 pts) Given two disjoint compact sets, A, B ⊂ ℝ^{n}, prove that there exists ϵ > 0 such that ||x − y|| > ϵ for any points x ∈ A and y ∈ B.

Solution: The product A×B of compact sets is compact, thus the function f: A×B → ℝ, f(x, y) = ||x − y|| (which is obviously continuous), takes it minimal value, m, on A × B. Note that m > 0, because f is strictly positive on A × B (we use here that A ∩ B ≠ ∅). If we choose 0 < ϵ < m, then the required condition is satisfied.

Problem 4. (10 pts) Represent X as a quotient of Y in the following cases

(1) X = S^{1}, Y = [0, 1]

(2) X is a torus, Y is a 2-disc

Solution: (1) The partition of [0, 1] contains {0, 1} and single-point-sets, {x}, x ∈ (0, 1). The correspondint quotient map can be defined as f: [0, 1] → S^{1}, f(t) = (cos(2πt), sin(2πt)).

(2) Consider the usual gluing scheme for a torus, but instead of a square take a disc (homeomorphic to the square).

Problem 5. (15 pts) Which ones of the following sets are – surfaces

– compact surfaces with boundary

– non-orientable surfaces

(1) S^{1} × [0, 1),

(2) S^{1} × S^{2},

(3) [0, 1] × [0, 1],

(4) D^{2} × S^{2},

(5) T^{2}#T^{2},

(6) M#T^{2} (M is a Möbius band),

(7) ℝP^{2}#T^{2},

(8) a Klein Bottle with a hole

(9) a punctured 2-sphere

(10) {(x, y, z) ∈ ℝ^{3} | y^{2} = x^{2} + z^{2}}

(11) {(x, y, z) ∈ ℝ^{3} | x = y^{2} − z^{3}}

Solution: S^{1} × S^{2} is a 3-dimensional and D^{2} × S^{2} is a 4-dimensional manifold (the latter has boundary). Example (10) is a cone and has a singularity at the origin, so it is a singular surface rather then an honest surface. The others are surfaces, among them (1), (3), (6), (8) are surfaces with boundary (if a hole in example (8) obtained by removing an open disk).

(3), (6), (8) represent compact surfaces with boundary (note that (5) and (7) are compact, but without boundary).

(6), (7) and (8) are non-orientable surfaces (all contain a Möbius band).

Problem 6. (20 pts) Assume that D_{1}, D_{2} ⊂ ℝ^{2} are topological 2-discs and h: ℝ^{2} → ℝ^{2} is a homeomorphism such that h(∂D_{1}) = (∂D_{2}). Prove that h(D_{1}) = D_{2}.

Problem 7. (20 pts) Give a gluing scheme that yields (1) an orientable surface of genus 2 with a hole,

(2) T^{2}#ℝP^{2}

(3) ℝP^{2} with 2 holes.

Problem 8. (20 pts) For the surfaces F sketched below (1) determine the Euler characteristic;

(2) point out which surfaces are homeomorphic, if there are any;

(3) determine the place of the surfaces in the classification list if they are closed (F_{g}, N_{k}) and point out the number of boundary components if they have any.

Explain your answer.