A Spectra comparison theorem and its applications

We give a sharp comparison between the spectra of two Riemannian manifolds (Y,g) and (X,g_0) under the following assumptions: (X,g_0) has bounded geometry, (Y,g) admits a continuous Gromov-Hausdorff {\epsilon}-approximation onto (X,g_0) of non zero absolute degree, and the volume of (Y,g) is almost smaller than the volume of (X,g_0). These assumption imply no restrictions on the local topology or geometry of (Y,g) in particular no curvature assumption is supposed or infered.