  After the significant discovery of the hole-doped nickelate compound
Nd$_{0.8}$Sr$_{0.2}$NiO$_2$, an analysis of the electronic structure, orbital
components, Fermi surfaces and band topology could be helpful to understand the
mechanism of its superconductivity. Based on the first-principles calculations,
we find that Ni $3d_{x^2-y^2}$ states contribute the largest Fermi surface.
$Ln~5d_{3z^2-r^2}$ states form an electron pocket at $\Gamma$, while $5d_{xy}$
states form a relatively bigger electron pocket at A. These Fermi surfaces and
symmetry characteristics can be reproduced by our two-band model, which
consists of two elementary band representations: $B_{1g}@1a~\oplus~A_{1g}@1b$.
We find that there is a band inversion near A, giving rise to a pair of Dirac
points along A--M below the Fermi level once including spin-orbit coupling.
Furthermore, we have performed the LDA+Gutzwiller calculations to treat the
strong correlation effect of Ni 3d orbitals. In particular, the bandwidth of
$3d_{x^2-y^2}$ has been renormalized largely. After the renormalization of the
correlated bands, the Ni $3d_{xy}$ states and the Dirac points become very
close to the Fermi level. Thus, a hole pocket at A could be introduced by hole
doping, which may be related to the observed sign change of Hall coefficient.
By introducing an additional Ni $3d_{xy}$ orbital, the hole-pocket band and the
band inversion can be captured in our modified model. Besides, the nontrivial
band topology in the ferromagnetic two-layer compound La$_3$Ni$_2$O$_6$ is
discussed and the band inversion is associated with Ni $3d_{x^2-y^2}$ and La
$5d_{xy}$ orbitals.
