METU Department of Mathematics Math 559 Low-dimensional Topology Spring 2009

CONTENT OF MIDTERM II AND EXERCISES

Content

(1) Braid monodromy of singularities, splitting of singularities and factorization of the braid monodromy, Hurwitz moves.

(2) Abstract branching coverings, existence and isomorphism of double branching coverings of surfaces.

(3) Mapping class group of surfaces and its relation to the Braid group; Dehn twists and their relations; mapping class groups of the simplest surfaces; the monodromy of surface singularity links (relation to the braid monodromy).

(4) Heegaard splittings and diagrams of 3-manifolds; relation to the Morse functions; the case of Lens spaces, stabilization.

(5) Dehn surgery on 3-manifolds, Kirby moves (handle slides, blowup move), slam-dunk.

(6) Attaching handles to B^{4}, handle slides, cancellation of handles, blowup.

Exercises

Problem 1. For the singularity of f(x, y) = x^{3} − y^{2} (1) find the braid monodromy by projecting to coordinate x;

(2) find the braid monodromy by projecting to coordinate y;

(3) describe the factorization of the monodromy in the cases 1–2 as we perturb f by adding to it εx (for a small ε): (what are the critical points after a perturbation and the corresponding monodromy

(4) give an example of a Hurwitz move in each of the cases (for projection to x and to y).

Problem 2.

(1) Describe the braid monodromy of f(x, y) = (x^{2} − y^{2})(4x^{2} − y^{2}) (project to axis y).

(2) Assume that f′ is a generic perturbation of f; find the number of critical points of f and explain their geometric meaning.

(3) How one can split off a singularity D_{4} from f ? Find the corresponding monodromy factorization.

(4) Is it possible to split of singularity E_{8} from f ? Explain.

(5) Give a few more examples of singularities which can generate f.

Problem 3.

(1) Describe all non-isomorphic double coverings over a disc with two punctures with three branching points: give the corresponding cutting set and explain how these covering can be distinguished.

(2) Find the topological type of the total space for each of these coverings.

Problem 4.

(1) Calculate (t_{a}t_{b})^{3} in the mapping class group of a torus S^{1} × S^{1}; where a and b represent the basic curves S^{1} × pt and pt × S^{1}.

(2) Represent a Dehn twist t_{2a+b} by a matrix in SL(2; ℤ).

Problem 5. Consider a surface singularity f^{2}(x, y) + z^{2}, where f(x, y) is like in Problem 2.

(1) Which surface is a page-fiber of the open book of this singularity ?

(2) Find the monodromy of this open book (Milnor fibration) representing it as a product of Dehn twists.

Problem 6.

(1) Sketch Heegaard diagrams of the lens spaces L(3, 1) and L(3, 2).

(2) Sketch a Heegaard diagram of genus 2 for S^{3}.

(3) What is the number of critical points of a Morse function corresponding to the Heegaard splittings in the above examples ? What are the indices of the critical points and in which order they do appear ?

Problem 7. Give a Kirby diagram for (1) the lens space L(43, 13) (with negative framings);

(2) the connected sum L(5, 2)#L(5, 3)#(S^{1} × S^{2});

(3) the link of the surface singularities D_{5}, E_{6}, and E_{7};

(4) the connected sum ℂP^{2}#\overline{ℂP^{2}}#(S^{2} × S^{2}) with a puncture.

Problem 8. Perform slam-dunk of the chain (2, 3, 5) twice: do it in three different ways. Explain why the three results are equivalent.