Topology Analysis of the Sloan Digital Sky Survey. I. Scale and Luminosity Dependence

We measure the topology of volume-limited galaxy samples selected from a parent sample of 314,050 galaxies in the Sloan Digital Sky Survey (SDSS), which is now complete enough to describe the fully three-dimensional topology and its dependence on galaxy properties. We compare the observed genus statistic G(Vf) to predictions for a Gaussian random field and to the genus measured for mock surveys constructed from new large-volume simulations of the (lambda)CDM cosmology. In this analysis we carefully examine the dependence of the observed genus statistic on the Gaussian smoothing scale RG from 3.5 to 11 h-1 Mpc and on the luminosity of galaxies over the range -22.50 < Mr < -18.5. The void multiplicity AV is less than unity at all smoothing scales. Because AV cannot become less than 1 through gravitational evolution, this result provides strong evidence for biased galaxy formation in low-density environments. We also find clear evidence of luminosity bias of topology within the volume-limited subsamples. The shift parameter (delta) v indicates that the genus of brighter galaxies shows a negative shift toward a "meatball" (i.e., cluster dominated) topology, while faint galaxies show a positive shift toward a "bubble" (i.e., void dominated) topology. The transition from negative to positive shift occurs approximately at the characteristic absolute magnitude Mr* = -20.4. Even in this analysis of the largest galaxy sample to date, we detect the influence of individual large-scale structures, as the shift parameter (delta) v and cluster multiplicity AC reflect (at ~3 (sigma)) the presence of the Sloan Great Wall and an X-shaped structure that runs for several hundred megaparsecs across the survey volume.