The Cut-off Covering Spectrum
We introduce the R cut-off covering spectrum and the cut-off covering spectrum of a complete
length space or Riemannian manifold. The spectra measure the sizes of localized holes in the
space and are defined using covering spaces called covers and R cut-off covers. They are
investigated using homotopies which are homotopies via grids whose squares are mapped into
balls of radius .
On locally compact spaces, we prove that these new spectra are subsets of the closure of the
length spectrum. We prove the R cut-off covering spectrum is almost continuous with respect
to the pointed Gromov-Hausdorff convergence of spaces and that the cut-off covering spectrum
is also relatively well behaved. This is not true of the covering spectrum defined in our earlier
work which was shown to be well behaved on compact spaces. We close by analyzing these
spectra on Riemannian manifolds with lower bounds on their sectional and Ricci curvature and
their limit spaces.
Complete length spaces and Riemannian manifolds are often studied using Gromov-Hausdorff con-
vergence and Gromov's compactness theorem. However, this convergence, reviewed in Section 5,