METU Department of Mathematics Math 541 Differential Topology Spring 2015 S.Finashin

Final, Part 2 (Take-home)

June 6, 31 points in 5 questions

Name and the student number:

Q1. (9 pts) Give a more simple description of manifolds obtained by the following constructions.

(1) The double of ℂP^{2} × D^{2} × D^{3}.

(2) The double of the Möbius band.

(3) A tubular neighborhood of S^{n} lying in a standard way in ℝ^{n+k}.

Q2. (9 pts) (a) Show that for any parallelizable manifold X its Euler characteristic χ(X) vanishes.

(b) Knowing that χ(X) = 0 deduce that the self-intersection index of the diagonal Δ_{X} ⊂ X × X is zero.

(c) Find the self-intersection index of the diagonal Δ_{F} ⊂ F × F where F is a closed oriented surface of genus 2.

Q3. (3 pts) Knowing that a vector field V in ℂP^{2} has only non-degenerate zeros, estimate from below the number of zeros.

Q4. (4 pts) Find the points in ℝ^{3} at which the map ℝ^{2} → ℝ^{3}, (x, y) ↦ (x, y^{2}, x^{2} + y^{2}) is not transversal to S^{2} = {x^{2} + y^{2} + z^{2} = 1}.

Q5. (6 pts) (a) Map f: X → Y is null-homotopic and g: X → Z is any map, dim X = dim Y + dim Z = n. Prove that the map X → Y × Z, x ↦ (f(x), g(x)) has degree 0.

(b) Show that any map S^{n} → S^{1} × X, where X is an (n − 1)-dimensional manifold, has degree 0.