Middle East Technical University Spring 2003 Math 710 Low Dimensional Topology

MIDTERM II (TAKE-HOME)

Problem 1. Let f: T^{2} → T^{2}, F = t_{a}^{2}t_{b}t_{a}^{−1}, where a and b are the standard meridian and longitude on a torus.

(1) Determine the matrix of the induced map f_{∗}: H_{1}(T^{2}) → H_{1}(T^{2}).

(2) Show that f is a product of right Dehn twists. Sketch the curves defining these Dehn twists.

(3) Determine the homology groups of the mapping tori M_{f} and M_{f^{2}}.

Problem 2. Let X_{c} denote the 3-manifold obtained from the standard Heegaard diagram of S^{3} of genus 1 after changing the gluing map of the tori by a Dehn twist t_{c}: T^{2} → T^{2}. Sketch a Kirby diagram of X_{c} for

(1) c = a

(2) c = a + b

(3) c = 2a + b where a, b are the meridian and longitude of the torus T^{2}.

Problem 3. Let X → ℂP^{2} be a double covering branched along a non-singular curves A ⊂ ℂP^{2} of degree 2k.

(1) find b_{2}^{±}(X), χ(X) and σ(X);

(2) determine c_{1}(X) (as a multiple of the “hyperplane class” h ⊂ H^{2}(X) coming from ℂP^{2}, and find c_{1}^{2}(X).

(3) determine the Hodge diamond of X;

(4) for which k the surface X is rational ? For which k it is Spin ?

Problem 4. Prove that a symplectic simply connected 4-manifold X must have b_{2}^{+}(X) odd.

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