Analytic equivariant coarse Novikov conjecture (injectivity of the Miscenko–Kasparov assembly map)

Establish rational injectivity of the Miscenko–Kasparov assembly map ν_X^Γ for every proper Γ-space X with equivariant bounded geometry, specifically the map ν_X^Γ: lim_{d,k→∞} K_*^Γ(P_{d,k}(X)) → K_*(C^*(X)^Γ), where P_{d,k}(X) are Milnor–Rips complexes modeling all free and proper Γ-spaces equivariantly coarsely equivalent to X.

Background

To handle groups with torsion and avoid set-theoretic issues, the authors introduce a Miscenko–Kasparov assembly map in the coarse equivariant setting by using Milnor–Rips complexes. The analytic equivariant coarse Novikov conjecture asserts rational injectivity of this map.

They prove rational injectivity under the hypothesis that X admits a Γ-equivariant coarse embedding into an admissible Hilbert–Hadamard space, but the general conjecture remains open; the statement is weaker than the equivariant coarse strong Novikov conjecture yet still yields significant applications (e.g., to positive scalar curvature).