  The topology of weak lensing fields is studied using the 2-dimensional genus
statistic. Simulated fields of the weak lensing convergence are used to focus
on the effect of nonlinear gravitional evolution and to model the statistical
errors expected in observational surveys. For large smoothing angles, the
topology is in agreement with the predictions from linear theory. On smoothing
angles smaller than 10', the genus curve shows the non-Gaussian signatures of
gravitational clustering and differs for open and flat cold dark matter models.
Forthcoming surveys with areas larger than 10 square degrees should have
adequate signal-to-noise to measure the non-Gaussian shape and the
$\Omega$-dependence of the genus statistic.
