Middle East Technical University Fall 2002 Math 744 Fundamental Techniques in Differential Topology

PROBLEM LIST VI (TAKE-HOME MIDTERM II)

Heegaard splitting and Kirby Calculus

Problem 1. Determine how the group H_{1}(M) of a 3-manifold M changes after surgery along a knot K ⊂ M with framing n. In particular

(1) consider the case of M = S^{3};

(2) show that if M is ℤHS then ±1-framed surgery along any knot K gives again ℤHS;

(3) show that H_{1}(M) does not change under ±1-framed surgery (knot K here is supposed to be null-homologous in M and framing is taken with respect to some Seifert surface for K);

(4) show that all Seifert surfaces for K in any 3-manifold M determine the same framing of K.

Problem 2. Knowing the critical points of Morse functions f: X → ℝ and g: Y → ℝ determine the ones for f ∘ p_{X} + g ∘ p_{Y} : X × Y → ℝ, where p_{X}, p_{Y} are the projections. In particular

(1) estimate the Heegaard genus of S^{1} × F, where F is an oriented closed surfaces of genus g;

(2) describe explicitly a Heegaard splitting of S^{1} × F of genus 2g + 1 (hint: note that F^{∘} ×[0, 1] is a handlebody of genus 2g, where F^{∘} is obtained from F by removing a small 2-disk).

(3) describe a similar Heegaard splitting for a mapping torus M_{f}, where f: F → F.

Problem 3. Show that the four framed links sketched below describe the same 3manifold.

Problem 4.

(1) If in a framed link L a component L_{0} is an unknot with framing 0 which links only one other component L_{1} geometrically once, then L_{0} ∪ L_{1} may be moved away from the link L without changing framings.

(2) Show that the two-component link L′ = L_{0} ∪ L_{1} defines S^{3}, observing that a handle slide of L_{1} along L_{0} preserves the link changing only the framing of L_{1} by ±2.

(3) Deduce from the above observations that M_{L} ≅ M_{L∖L′}, that is, if we remove L_{0} and L_{1} from L than the 3-manifold described by the link will not change.

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