  In this paper, we consider the eigenvalue problem for Hodge-Laplacian on a
Riemannian manifold $M$ isometrically immersed into another Riemannian manifold
$\bar M$ for arbitrary codimension. We first assume the pull back
Weitzenb\"{o}ck operator (defined in Section 2) of $\bar M$ bounded from below,
and obtain an extrinsic lower bound for the first eigenvalue of
Hodge-Laplacian. As applications, we obtain some rigidity results and a
homology sphere theorem. Second, when the pull back Weitzenb\"{o}ck operator of
$\bar M$ bounded from both sides, we give a lower bound of the first eigenvalue
by the Ricci curvature of $M$ and some extrinsic geometry. As a consequence, we
prove a weak Ejiri type theorem, that is, if the Ricci curvature bounded from
below pointwisely by a function of the norm square of the mean curvature
vector, then $M$ is a homology sphere. In the end, we give an example to show
that all the eigenvalue estimates and homology sphere theorems are optimal when
$\bar M$ has constant curvature.
