Contractibility of the Rips complexes of Integer lattices via local domination

Žiga Virk

doi:10.1090/tran/9308

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arXiv · 2405.09134

Contractibility of the Rips complexes of Integer lattices via local domination

Žiga Virk

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Abstract

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 $n$

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Žiga Virk. 2024-05-15. Contractibility of the Rips complexes of Integer lattices via local domination. https://doi.org/10.1090/tran%2F9308

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