  In this paper, we consider immersed two-sided minimal hypersurfaces in
$\mathbb{R}^n$ with finite total curvature. We prove that the sum of the Morse
index and the nullity of the Jacobi operator is bounded from below by a linear
function of the number of ends and the first Betti number of the hypersurface.
When $n=4$, we are able to drop the nullity term by a careful study for the
rigidity case. Our result is the first effective generalization of Li-Wang.
Using our index estimates and ideas from the recent work of
Chodosh-Ketover-Maximo, we prove compactness and finiteness results of minimal
hypersurfaces in $\mathbb{R}^4$ with finite index.

Differential Geometry