  We trace the connectivity of the cosmic web as defined by haloes in the
Planck-Millennium simulation using a persistence and Betti curve analysis. We
normalise clustering up to the second-order correlation function, and use our
systematic topological analysis to correlate local information and properties
of haloes with their multi-scale geometrical environment of the cosmic web
(elongated filamentary bridges and sheetlike walls). We capture the multi-scale
topology traced by the halo distribution through filtrations of the
corresponding Delaunay tessellation. The resulting nested $\textit{alpha
shapes}$ are sensitive to the local density, perfectly outline the local
geometry, and contain the complete information on the multi-scale topology. We
find a remarkable linear relationship between halo masses and topology: haloes
of different mass trace environments with different topological signature. This
is $\textit{topological bias}$, an environmental structure bias independent of
the halo clustering bias associated with the two-point correlation function.
This mass-dependent linear scaling relation allows us to take clustering into
account and determine the overall connectivity from a limited sample of
galaxies. The presence of topological bias has major implications for the study
of voids and filaments in the observed distribution of galaxies. The
(infra)structure and shape of these key cosmic web components will strongly
depend on the underlying galaxy sample. Their use as cosmological probes, with
their properties influenced by cosmological parameters, will have to account
for the subtleties of topological bias. This is of particular relevance with
the large upcoming galaxy surveys such as DESI, Euclid, and the Vera Rubin
telescope surveys.
