  We present measurements of the genus topology of luminous red galaxies (LRGs)
from the Sloan Digital Sky Survey (SDSS) Data Release 7 catalog, with
unprecedented statistical significance. To estimate the uncertainties in the
measured genus, we construct 81 mock SDSS LRG surveys along the past light cone
from the Horizon Run 3, one of the largest N-body simulations to date that
evolved 7210^3 particles in a 10815 Mpc/h size box. After carefully modeling
and removing all known systematic effects due to finite pixel size, survey
boundary, radial and angular selection functions, shot noise and galaxy
biasing, we find the observed genus amplitude to reach 272 at 22 Mpc/h
smoothing scale with an uncertainty of 4.2%; the estimated error fully
incorporates cosmic variance. This is the most accurate constraint of the genus
amplitude to date, which significantly improves on our previous results. In
particular, the shape of the genus curve agrees very well with the mean
topology of the SDSS LRG mock surveys in the LCDM universe. However, comparison
with simulations also shows small deviations of the observed genus curve from
the theoretical expectation for Gaussian initial conditions. While these
discrepancies are mainly driven by known systematic effects such as those of
shot noise and redshift-space distortions, they do contain important
cosmological information on the physical effects connected with galaxy
formation, gravitational evolution and primordial non-Gaussianity. We address
here the key role played by systematics on the genus curve, and show how to
accurately correct for their effects to recover the topology of the underlying
matter. In a forthcoming paper, we provide an interpretation of those
deviations in the context of the local model of non-Gaussianity.
