METU Department of Mathematics Math 543 Low-dimensional Topology Fall 2013 S.Finashin

Midterm 2 Part 1 (Take-home)

due to December 8, 13:30:, 50 points in 7 problems

Name and the student number:

Q1. (10 pts) Let F_{g} denote a closed oriented surfaces of genus g, and F_{g,n} is such a surface with n punctured (i.e., with n discs removed).

(a) Assume that F_{g} → F_{3} is a triple covering with 5 ramification points, 2 of which are simple and the other 3 have ramification index 3. Find g.

(b) Prove that there cannot be a branched covering F_{g} → F_{g′}, if g < g′.

(c) Find g and n for a threefold covering F_{g,n} → D^{2} with 5 simple branch points (if the answer is not unique, explain why, and give several possible answers).

Q2. (10 pts) Assume that a 5-fold branched covering f: F → ℂ has branch points b_{1}, b_{2}, b_{3} ∈ ℂ and the corresponding Hurwitz monodromy factorization in S_{5} is μ_{1} = (123)(45), μ_{2} = (13)(24), and μ_{3} = (25).

(a) What is the ramification profile at each of the branch points b_{i} ?

(b) How many points are contained in the inverse images f^{−1}(b_{i}) for i = 1, 2, 3 ?

(c) How many ramification points are contained in F ?

(d) Find the monodromy of f at infinity.

(e) If we extend f at infinity to a map of closed surfaces f̂: F̂ → ℂP^{2}, then how many points will be in f̂^{−1}([∞]) ? How many of them are ramification points and what are their ramification indices ?

Q3. (12 pts) Consider a rational function, f(z) = \frac{f_{1}(z)}{f_{2}(z)}, where f_{1}(z), f_{2}(z) are polynomials, and extend it through infinity to F : ℂP^{1} → ℂP^{1} as follows

F ([z : w]) = [f_{1}(\frac{z}{w}) : f_{2}(\frac{z}{w})].

(Extension here means that F ([z : 1]) = [f(z) : 1].)

(a) In the case f_{1}(z) = z^{3} and f_{2}(z) = z^{2}+1, show that F is a threefold branched covering.

(b) Can you extend your arguments to any pair of polynomials ? What will be the degree of the branched covering ? (A Hint: what happens if f_{1} and f_{2} have a common root ?)

(c) How to express in terms of f_{1} and f_{2} that f (and thus, F) has a branch point of ramification index 3 at z = 0 and a simple branch point at z = 1 ?

(d) What you can say about the number n of simple branch points of F if it is a 5-fold branched covering, which has in addition to the simple branch points just one branch point of ramification index 3 ?

Q4. (4 pts) Let m and ℓ be a meridian and a longitude of a torus T^{2}, and pm+qℓ denote a curve representing the homology class (p, q) ∈ ℂ^{2} = H_{1}(T^{2}).

(a) Sketch a curve m + ℓ on T^{2} and its images under the Dehn twists t_{a} and t_{b}.

(b) Present the above images t_{a}(m + ℓ) and t_{b}(m + ℓ) in the form pm + qℓ.

Q5. (5 pts) Let f: T^{2} → T^{2} be an orientation preserving homeomorphism. (a) Is it possible that f(m) = 2m + 3ℓ, or f(m) = 2m + 4ℓ ? Explain.

(b) Knowing that f(m) = m + 2ℓ, find p and q such that f(ℓ) = pm + qℓ. Is the answer unique ? (A Hint: use that f preserves the intersection form in H_{1}(T^{2}) ).

Q6. (3 pts) Justify the relations T_{a}T_{b}T_{a} = T_{b}T_{a}T_{b} and (T_{a}T_{b})^{6} = I for the matrices T_{a}, T_{b} representing the action of the Dehn twists t_{a}, t_{b} along curves a, b on a torus T^{2}, giving a basis in H_{1}(T^{2}) = ℂ^{2}.

Q7. (6 pts) Sketch Heegaard diagrams of genus 2 that represent

(a) S^{3}; (b) S^{1} × S^{2}; (c) S^{1} × S^{2}#S^{1} × S^{2}; (c) L(3, 1).