Lefschetz Fibrations and disk bundles When reading chaptes 7 of Akbulut's book about $4-$manifolds, he describes a handle decomposition for a manifold given a Lefschetz fibration over $S^2$. The idea is to extend the preimage of a disk with no critical points to a disk containing the critical points via neighborhoods of arcs connecting a fixed point with the critical values (each corresponding to a $2-$handle). The Kirby diagram obtained from this construction, how far is to be "minimal" (least number of handles)? Is the same construction above applicable for Lefschetz fibrations with image positive genus surface (surface bundle over some surface with two handles attached)? 
Can someone recommend a reference for this topic? 
 A: In general I think it is hard to find a "minimal" handle decomposition: the problem could be for example that if you have torsion in $H_k$, then you would need some $(k+1)$-handle which is not "visible" as a generator inside $H_{k+1}(-;\mathbb{Z})$. I guess it is the same for any CW-decomposition: what is the minimal number of cells you need to realize a space $X$? You need to look at all the possible $H_*(X;\mathbb{Z}/n\mathbb{Z})$ for varying $n$ in order to detect the cells which produce torsion.
In theory, the algorithm for a handle decomposition works the same over any surface: you can remove the disks containing the images of the critical points and get a surface bundle over a surface-with-boundary: you know how many 1-handles you have to attach, but now you get extra 2-handles coming as product of 1-handles in the fiber and 1-handles in the base. The last step would be attaching more 2-handles according to the number and position of the vanishing cycles.
You can find a good review of handle decomposition of a Lefschetz fibration (over the disk) at page 39 of Ozbagci's "Lectures on the topology of symplectic fillings of contact 3-manifolds".
