  Let $G_{\infty}=(C_m^d)_{\infty}$ denote the graph whose set of vertices is
$\{1,..., m\}^d$, where two distinct vertices are adjacent iff they are either
equal or adjacent in $C_m$ in each coordinate. Let $G_{1}=(C_m^d)_1$ denote the
graph on the same set of vertices in which two vertices are adjacent iff 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 Bollob\'as, 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.
