Upper bounds for the minimal number of singular fibers in a Lefschetz   fibration over the torus
  In this paper, we give some relations in the mapping class groups of oriented closed surfaces in the form that a product of a small number of right hand Dehn twists is equal to a single commutator. Consequently, we find upper bounds for the minimal number of singular fibers in a Lefschetz fibration over the torus. 
Noriyuki Hamada
math.GT