Monodromies of rational elliptic surfaces and extremal elliptic $K3$ surfaces

We give a systematic method to calculate some homological data from the global monodromy of a topological elliptic surface. We apply this method to the cases 1) the transcendental lattice of an extremal elliptic K3 surface, 2) the torsion part of Mordell-Weil group of a general elliptic surface, and 3) the Mordell-Weil lattice of a rational elliptic surface. We present possible list of global monodromies of rational elliptic surface and extremal elliptic $K3$ surface which reproduce to the classification obtained by Oguiso-Shioda and Shimada-Zhang respectively.