The coarse Novikov conjecture for discrete metric spaces

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Let XX be a discrete metric space with bounded geometry. For each d>0d>0, let Pd(X)P_d(X) be its Rips complex, and let CL∗(Pd(X))C^*_L(P_d(X)) and C∗(X)C^*(X) denote its localization and Roe algebras. The evaluation map is

ev ⁣:lim⁡d→∞CL∗(Pd(X))→C∗(X).ev\colon \lim_{d\to\infty} C^*_L(P_d(X))\to C^*(X).

Coarse Novikov conjecture. The induced map

ev∗ ⁣:lim⁡d→∞K∗(CL∗(Pd(X)))→K∗(C∗(X))ev_*\colon \lim_{d\to\infty}K_*( C^*_L(P_d(X)))\to K_*( C^*(X))

is injective. This conjecture asserts injectivity of the coarse assembly map and is the coarse analogue of the Strong Novikov Conjecture. Its resolution is not supplied in the source.

References

Primary source

Jinmin Wang and Bo Zhu, “Sharp bottom spectrum and scalar curvature rigidity”, arXiv:2408.08245 (2026).

Additional references

2 papers in this index state this conjecture (2022–2024). The statement above is taken from the most recent of them; the others are arXiv:2207.04193.

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