An Algorithmically Unsolvable Problem in Analysis
A. Lenard, J. Stillwell · Proceedings of the American Mathematical Society · 1983
The decision problem of distinguishing between the cases when the Laplace-Beltrami operator on the covering space of a compact manifold has 0 in its spectrum or is bounded away from 0 is algorithmically unsolvable in any class of manifolds that includes all $4$-dimensional ones. The proof depends on a result of Brooks connecting the spectrum with the amenability of the fundamental group.