Contractibility of the Rips complexes of Integer lattices via local domination

We prove that for each positive integer n, the Rips complexes of the n-dimensional integer lattice in the d₁ metric (i.e., the Manhattan metric, also called the natural word metric in the Cayley graph) are contractible at scales above n²(2n-1)², 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