Eigenvalue estimates and differential form Laplacians on Alexandrov spaces

We give upper bounds on the eigenvalues of the differential form Laplacian on a compact Riemannian manifold. The proof uses Alexandrov spaces with curvature bounded below. We also construct differential form Laplacians on Alexandrov spaces. Under a local biLipschitz assumption on the Alexandrov space, which is conjecturally always satisfied, we show that the differential form Laplacian has a compact resolvent. We identify its kernel with an intersection homology group.