METU Department of Mathematics Math 534 Low-dimensional Topology Spring 2009

FINAL

Problem 1. Sketch a Kirby diagram of the following 4-manifolds. (1) ℂP^{2} blown up twice,;

(2) S^{2} × S^{2};

(3) S^{1} × S^{3};

(4) Mazur manifold; explain what makes it a remarkable example ?

(5) plumbing 4-manifold for the graph E_{6};

(6) a tubular neighborhood of a non-singular complex conic curve in ℂP^{2}.

Problem 2. Sketch a Kirby diagram of the following 3-manifolds. (1) S^{1} × S^{2}#ℝP^{3};

(2) 3-torus;

(3) L(8, 3);

(4) Poincare sphere;

(5) Σ(3, 4, 5).

Problem 3. Find the topological type (give a sketch) and self-intersection index of the following complex algebraic curves.

(1) A non-singular curve of degree 5 in ℂP^{2}.

(2) A non-singular curve of degree (4, 2) in ℂP^{1} × ℂP^{1}.

(3) A curve of degree 4 in ℂP^{2} with two nodal singularities.

(4) A curve of degree (3, 4) in ℂP^{1} × ℂP^{1} with two cuspidal singularities.

Problem 4. Answer the following questions with YES, NO, or NOT KNOWN. Explain by citing relevant theorems, examples, or problems.

(1) Is there a 3-manifold homeomorphic but not diffeomorphic to ℝP^{3} ?

(2) Is there a simply connected 3-manifold not homeomorphic to S^{3} ?

(3) Is there a four-manifold which does not admit a smooth structure ?

(4) Is there a four-manifold with the intersection form E_{8} ?

(5) Is there a smooth four-manifold with the intersection form E_{8} + E_{8} ?

(6) Is there a four-manifold with the intersection form U, which is not homeomorphic to S^{2} × S^{2} ?

(7) Is there a smooth four manifold with the intersection form U which is not diffeomorphic to S^{2} × S^{2} ?