  We prove that for each positive integer $n$, the Rips complexes of the
$n$-dimensional integer lattice in the $d_1$ metric (i.e., the Manhattan
metric, also called the natural word metric in the Cayley graph) are
contractible at scales above $n^2(2n-1)$, with the bounds arising from the
Jung's constants. We introduce a new concept of locally dominated vertices in a
simplicial complex, upon which our proof strategy is based. This allows us to
deduce the contractibility of the Rips complexes from a local geometric
condition called local crushing. In the case of the integer lattices in
dimension $n$ and a fixed scale $r$, this condition entails the comparison of
finitely many distances to conclude that the corresponding Rips complex is
contractible. In particular, we are able to verify that for $n=1,2,3$, the Rips
complex of the $n$-dimensional integer lattice at scale greater or equal to $n$
is contractible. We conjecture that the same proof strategy can be used to
extend this result to all dimensions $n$
