Summary: Let \(G_\infty=(C^d_m)_\infty\) denote the graph whose set of vertices is \(\{0,\dots,m-1\}^d\), where two distinct vertices are adjacent if and only if they are either equal or adjacent in the \(m\)-cycle \(C_m\) in each coordinate. Let \(G_1=(C^d_m)_1\) denote the graph on the same set of vertices in which two vertices are adjacent if and only if they are adjacent in one coordinate in \(C_m\) and equal in all others. Both graphs can be viewed as graphs of the \(d\)-dimensional torus. We prove that one can delete \(O(\sqrt{d}m^{d-1})\) vertices of \(G_1\) so that no topologically nontrivial cycles remain. This improves an \(O(d^{\log_2 (3/2)}m^{d-1})\) estimate of Bollobás, Kindler, Leader and O’Donnell. We also give a short proof of a result implicit in a recent paper of Raz: one can delete an \(O(\sqrt{d}/m)\) fraction of the edges of \(G_\infty\) so that no topologically nontrivial cycles remain in this graph. Our technique also yields a short proof of a recent result of Kindler, O’Donnell, Rao and Wigderson; there is a subset of the continuous \(d\)-dimensional torus of surface area \(O(\sqrt{d})\) that intersects all nontrivial cycles. All proofs are based on the same general idea: the consideration of random shifts of a body with small boundary and no nontrivial cycles, whose existence is proved by applying the isoperimetric inequality of Cheeger or its vertex or edge discrete analogues.