Intersection of Curves and Crossing Number of C_m × C_n on Surfaces

Let (K 1 , 2 ) be two families of closed curves on a surface S, such that |= m,|K 2 |=n,m #m#n, each curve in 1 intersects each curve in 2 , and no point of is covered three times. When is the plane, the projective plane, or the Klein bottle, we prove that the total number of intersections in 2 is at least 10mn/9, 12mn/11, and 10 -13 m , respectively. Moreover, when m is close to n, the constants are improved. For instance, the constant for the plane, 10/9, is improved to 8/5, for n 1)/4. Consequently, we prove lower bounds on the crossing number of the Cartesian product of two cycles, in the plane, projective plane, and the Klein bottle. All lower bounds are within small multiplicative factors from easily derived upper bounds. No general lower bound has been previously known, even on the plane. 1