Contractibility of the Rips complexes of Integer lattices via local domination

We prove that for each positive integer n n, the Rips complexes of the n n -dimensional integer lattice in the d 1 d₁ metric (i. e. , the Manhattan metric, also called the natural word metric in the Cayley graph) are contractible at scales above n 2 (2 n − 1) n² (2n-1), with the bounds arising from the Jung 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 n and a fixed scale r 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 n=1, 2, 3, the Rips complex of the n n -dimensional integer lattice at scale greater or equal to n n is contractible. We conjecture that the same proof strategy can be used to extend this result to all dimensions n n.