Middle East Technical University Low-dimensional Topology Math 710 Spring 2005

Midterm II

(take-home)

Problem 1. Consider knots K: a) figure-8 knot; b) torus knot K_{2,5} and consider the double cover M_{K} → S^{3} branched along K.

(1) Present M_{K} as an open book;

(2) give a Heegaard diagram of M_{K};

(3) give a Kirby diagram of M_{K};

(4) determine H_{1}(M_{K}).

Problem 2. Let X_{d} → ℂP^{1} × ℂP^{1} denotes the double covering branched along a non-singular curve of degree (6, 2d) (such complex surfaces are called Horikava surfaces). Determine

(1) χ(X_{d}),

(2) b_{i}(X_{d}), for all i,

(3) σ(X_{d}), b_{2}^{±}(X_{d}),

(4) p_{g}(X_{d}), c_{1}^{2}(X),

(5) h^{i,j}(X_{d})

Problem 3. Let C_{m,n} be the space of all curve of degree (m, n) in X = ℂP^{1} ×ℂP^{1}. (1) Determine the genus of non-singular curves A ∈ C_{m,n}

(2) What is the dimension of C_{m,n} ?

(3) Find the dimension of the subvariety R_{m.n} ⊂ C_{m,n} formed by rational curves.

(4) Which minimal number of points one need to fix in X, so that there will be only finitely many curves A ∈ R_{m,n} passing through these points ?

Problem 4. Show the scheme of resolution for the singularities {f_{i}(x, y) = 0}, where f_{1}(x, y) = x^{4} + y^{3}, f_{2}(x, y) = (x^{3} + y^{2})y, f_{3}(x, y) = x^{5} + y^{3}. Justify that the resolution graphs for the surface singularities {f_{i} + z^{2} = 0}, i = 1, 2, 3 are E_{6}, E_{7}, E_{8}.

Problem 5. Consider a plumbing 4-manifold X = P (Γ), where Γ is the weighted tree shown below.

(1) minimize Γ by contraction and sketch the new graph Γ_{1} that you obtain.

(2) What is the relation between the manifolds X and X_{1} = P (Γ_{1}) ?

(3) Determine the matrix of the intersection form of X_{1}. Is it definite ? Is it even ? (explain)

(4) What is the order of H_{1}(∂X_{1}) ? Determine this group.

(5) Can Γ_{1} and Γ be resolution graphs of some singularities ?

(6) Sketch a Kirby diagram representing X and X_{1}.

(7) Assuming that X_{1} is the resolution graph of a singularity, evaluate c_{1}(X_{1}) on the components of the exceptional divisor E.

(8) Represent c_{1}(X_{1}) as a linear combination of the components of E.