Economical Elimination of Cycles in the Torus

Let m > 2 be an integer, let C2m denote the cycle of length 2m on the set of vertices [−m,m) = {−m,−m+ 1, . . . ,m− 2,m− 1}, and let G = G(m, d) denote the graph on the set of vertices [−m,m), in which two vertices are adjacent iff they are adjacent in one coordinate of C2m and equal in all others. This graph can be viewed as the graph of the d-dimensional torus. We prove that one can delete a fraction of at most O( log d m ) of the vertices of G so that no topologically nontrivial cycles remain. This is tight up to the log d factor and improves earlier estimates by various researchers.