We study the two-dimensional topology of the galactic distribution when projected onto two-dimensional spherical shells. Using the latest Horizon Run 4 simulation data, we construct the genus of the two-dimensional field and consider how this statistic is affected by late-time nonlinear effects -- principally gravitational collapse and redshift space distortion (RSD). We also consider systematic and numerical artifacts such as shot noise, galaxy bias, and finite pixel effects. We model the systematics using a Hermite polynomial expansion and perform a comprehensive analysis of known effects on the two-dimensional genus, with a view toward using the statistic for cosmological parameter estimation. We find that the finite pixel effect is dominated by an amplitude drop and can be made less than $1%$ by adopting pixels smaller than $1/3$ of the angular smoothing length. Nonlinear gravitational evolution introduces time-dependent coefficients of the zeroth, first, and second Hermite polynomials, but the genus amplitude changes by less than $1%$ between $z=1$ and $z=0$ for smoothing scales $R_{rm G} > 9 {rm Mpc/h}$. Non-zero terms are measured up to third order in the Hermite polynomial expansion when studying RSD. Differences in shapes of the genus curves in real and redshift space are small when we adopt thick redshift shells, but the amplitude change remains a significant $sim {cal O}(10%)$ effect. The combined effects of galaxy biasing and shot noise produce systematic effects up to the second Hermite polynomial. It is shown that, when sampling, the use of galaxy mass cuts significantly reduces the effect of shot noise relative to random sampling.
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