  We report a study of the Fermi surface of the chiral semimetal CoSi and its
relationship to a network of multifold topological crossing points,Weyl points,
and topological nodal planes in the electronic band structure. Combining
quantum oscillations in the Hall resistivity, magnetization, and torque
magnetization with ab initio electronic structure calculations, we identify two
groups of Fermi-surface sheets, one centered at the R point and the other
centered at the $\Gamma$ point. The presence of topological nodal planes at the
Brillouin zone boundary enforces topological protectorates on the Fermi-surface
sheets centered at the R point. In addition, Weyl points exist close to the
Fermi-surface sheets centered at the R and the $\Gamma$ points. In contrast,
topological crossing points at the R point and the $\Gamma$ point, which have
been advertised to feature exceptionally large Chern numbers, are located at a
larger distance to the Fermi level. Representing a unique example in which the
multitude of topological band crossings has been shown to form a complex
network, our observations in CoSi highlight the need for detailed numerical
calculations of the Berry curvature at the Fermi level, regardless of the
putative existence and the possible character of topological band crossings in
the band structure.
