  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.
