METU Department of Mathematics Math 543 Low-dimensional Topology Fall 2013 S.Finashin

Final, Part 2

January 7 2014, 17:40 – 19:00, 42 points in 6 problems

Name and the student number:

Q1. (6 pts) Which of the following matrices may describe the intersection form of some compact 4-manifolds ? Which ones may describe the intersection of some closed 4-manifolds ? In which cases the intersection form is even and in which

cases is odd ? (Explain.) ( \begin{matrix}2 & 3 \\ 3 & 4 \end{matrix} ), ( \begin{matrix}2 & 3 \\ 5 & 8 \end{matrix} ), ( \begin{matrix}0 & 2 \\ 2 & 0 \end{matrix} ), ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ), ( \begin{matrix}2 & 0 \\ 0 & −1 \end{matrix} ).

Solution: Matrices ( \begin{matrix}2 & 3 \\ 5 & 8 \end{matrix} ) and ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) and not symmetric and thus, can-

not describe the intersection form of a compact 4-manifold. The other three matrices do describe.

Matrix ( \begin{matrix}2 & 3 \\ 3 & 4 \end{matrix} ) describes an intersection form of a closed 4-manifold because it is

symmetric and unimodular. The other are either not symmetric, or not unimodular and so, do not describe.

The intersection form is even (containing even numbers on the diagonal) in the

cases of ( \begin{matrix}2 & 3 \\ 3 & 4 \end{matrix} ) and ( \begin{matrix}0 & 2 \\ 2 & 0 \end{matrix} ). It is odd in the cases of ( \begin{matrix}2 & 0 \\ 0 & −1 \end{matrix} ). (In the other

two cases the matrices do not represent intersection forms.)

Q2. (6 pts) Which of the following 4-manifolds do not admit an almost complex structure ? Which ones do not admit a symplectic structure ? (Explain.)

(a) \overline{ℂP^{2}},

Solution: b_{2}^{+}+b_{1} = 0 is even, so, it does not admit an almost complex structure, and thus, does not admit a symplectic structure (since every symplectic manifold admits an almost complex structure).

(b) ℂP^{2}#ℂP^{2}#ℂP^{2},

Solution: b_{2}^{+} + b_{1} = 3 is odd, and moreover, 3σ + 2χ = 9 + 10 = 19 is a square of c_{1} = (3, 3, 1) ∈ ℂ^{3} = H_{2}(ℂP^{2}#ℂP^{2}#ℂP^{2}), c_{1}^{2} = 9 + 9 + 1 = 19.

So, it admits an almost complex structure. But it does not admit a symplectic structure, since it is a connected sum of

manifolds with b_{2}^{+} > 0.

(c) ℂP^{2}#\overline{ℂP^{2}}#\overline{ℂP^{2}}#\overline{ℂP^{2}},

Solution: ℂP^{2} is an algebraic surface and blowing up an algebraic surface yields an algebraic surface.

Thus, ℂP^{2}#\overline{ℂP^{2}}#\overline{ℂP^{2}}#\overline{ℂP^{2}} is an algebraic surface and therefore a symplectic manifold.

(d) (S^{2} × S^{2})#(S^{2} × S^{2}).

Solution: Its intersection form U ⊕ U has positive (and negative) inertia index 1 + 1 = 2.

It is simply connected as a connected sum of simply connected manifolds. Thus, b_{2}^{+} + b_{1} = 2 + 0 = 2 which is even, and so, this manifold has no almost

complex structure, and therefore, has no symplectic structure.

(e) (S^{3} × S^{1}).

Solution: It has a complex structure, and thus, admits an almost complex structure. (One can also see that b_{2}^{+} + b_{1} = 0 + 1 is odd, and 3σ + 2χ = 0 + 0 is the square of c_{1} = 0 ∈ H_{2}(S^{3} × S^{1}) = 0.)

But it is not symplectic, because b_{2}^{+} = 0, whereas the homology class [ω] represented by the symplectic form ω should have positive square.

Q3. (8 pts) (a) Given a Lefschetz fibration X → S^{2} with a fiber of genus 2, find the number of singular fibers, if it is known that X is simply connected and b_{2}(X) = n.

(b) What can you say about possible values of n, if all the fibers are irreducible ?

Q4. (6 pts) (a) How many base-points should have a (generic) pencil of quintics in ℂP^{2} ?

(b) How many singular fibers should it contain ?

Q5. (10 pts) Give an example of a monodromy factorization for a Lefschetz fibration (a) on the manifold E(3),

(b) on some manifold X, with a fiber of genus 3.

Q6. (6 pts) Determine the Thurston-Bennequin framing and the rotation number of the following Legendrian knots.