  A novel method to characterize the topology of the early-universe
intergalactic medium during the epoch of cosmic reionization is presented. The
21-cm radiation background from high redshift is analyzed through the
calculation of the 2-dimensional (2D) genus. The radiative transfer of
hydrogen-ionizing photons and ionization-rate equations are calculated in a
suite of numerical simulations under various input parameters. The 2D genus is
calculated from the mock 21-cm images of the high-redshift Universe. We
construct the 2D genus curve by varying the threshold differential brightness
temperature, and compare this to the 2D genus curve of the underlying density
field. We find that (1) the 2D genus curve reflects the evolutionary track of
cosmic reionization and (2) the 2D genus curve can discriminate between certain
reionization scenarios and thus indirectly probe the properties of
radiation-sources. Choosing the right beam shape of a radio antenna is crucial
for this analysis. To this end, the Square Kilometre Array (SKA) is found to be
a suitable apparatus for this analysis in terms of sensitivity, even though
some deterioration of the data for this purpose is unavoidable under the
planned size of the antenna core.
