Michael Freedman, Matthew Hastings

We give a procedure for “reverse engineering" a closed, simply connected, Riemannian manifold with bounded local geometry from a sparse chain complex over \({\mathbb {Z}}\). Applying this procedure to chain complexes obtained by “lifting" recently developed quantum codes, which correspond to chain complexes over \({\mathbb {Z}}_2\), we construct the first examples of power law \({\mathbb {Z}}_2\) systolic freedom. As a result that may be of independent interest in graph theory, we give an efficient randomized algorithm to construct a weakly fundamental cycle basis for a graph, such that each edge appears only polylogarithmically times in the basis. We use this result to trivialize the fundamental group of the manifold we construct.