  We study the topology of the matter density field in two dimensional slices,
and consider how we can use the amplitude $A$ of the genus for cosmological
parameter estimation. Using the latest Horizon Run 4 simulation data, we
calculate the genus of the smoothed density field constructed from lightcone
mock galaxy catalogs. Information can be extracted from the amplitude of the
genus by considering both its redshift evolution and magnitude. The constancy
of the genus amplitude with redshift can be used as a standard population, from
which we derive constraints on the equation of state of dark energy $w_{\rm
de}$ - by measuring $A$ at $z \sim 0.1$ and $z \sim 1$, we can place an order
$\Delta w_{\rm de} \sim {\cal O}(15\%)$ constraint on $w_{\rm de}$. By
comparing $A$ to its Gaussian expectation value we can potentially derive an
additional stringent constraint on the matter density $\Delta \Omega_{\rm mat}
\sim 0.01$. We discuss the primary sources of contamination associated with the
two measurements - redshift space distortion and shot noise. With accurate
knowledge of galaxy bias, we can successfully remove the effect of redshift
space distortion, and the combined effect of shot noise and non-linear
gravitational evolution is suppressed by smoothing over suitably large scales
$R_{\rm G} \ge 15 {\rm Mpc}/h$. Without knowledge of the bias, we discuss how
joint measurements of the two and three dimensional genus can be used to
constrain the growth factor $\beta = f/b$. The method can be applied optimally
to redshift slices of a galaxy distribution generated using the drop-off
technique.
