Eliminating cycles in the discrete torus

In this paper we consider the following question: how many vertices of the discrete torus must be deleted so that no topologically nontrivial cycles remain? We look at two different edge structures for the discrete torus. For (ℤ _m ^d ) ₁, where two vertices in ℤ _m are connected if their ℓ ₁ distance is 1, we show a nontrivial upper bound of d ^log ₂(3/2)m ^d-1 d ^0.6m ^d-1 on the number of vertices that must be deleted. For (ℤ _m ^d ) _∞, where two vertices are connected if their ℓ _∞ distance is 1, Saks et al. (Combinatorica 24(3):525-530, 2004) already gave a nearly tight lower bound of d(m-1) ^d-1 using arguments involving linear algebra. We give a more elementary proof which improves the bound to m ^d -(m-1) ^d , which is precisely tight.