Coarse Novikov conjecture with coefficients

Determine whether for every metric space X with bounded geometry and every coarse X-algebra (X, A), the twisted assembly map mu_{(X,A)}: lim_{d→∞} K_*(C^*_{L,(X,A)}(P_d(X), A)) → K_*(C^*_{(X,A)}(X, A)) is injective.

Background

The coarse Novikov conjecture concerns the injectivity of the coarse assembly map and is a fundamental consequence of the coarse Baum–Connes framework. In this paper, the authors formulate a twisted version with coefficients in coarse X-algebras, paralleling their twisted Baum–Connes setup.

They further establish partial results: if X has a coarse Z-fibration structure with bounded geometry, the twisted coarse Novikov conjecture holds for X provided the base space Y satisfies the twisted coarse Novikov conjecture and the fiber Z satisfies the twisted coarse Baum–Connes conjecture. This provides supporting evidence and a pathway toward the general conjecture with coefficients.