Rational strong Novikov conjecture

Let $\Gamma$ be a discrete group, and for each finite symmetric subset $F\subseteq\Gamma$ containing the identity, let $P_F(\Gamma)$ be the corresponding Rips complex. Let $e$ be the evaluation map from the equivariant localization algebra to the equivariant Roe algebra.

Strong Novikov conjecture. The map $e$ induces an injection

$$e_*:\varinjlim_F K_*(C^*_L(P_F(\Gamma))^\Gamma)\longrightarrow\varinjlim_F K_*(C^*(P_F(\Gamma))^\Gamma),$$

where the limit is over all finite symmetric subsets $F$ of $\Gamma$ containing the identity. In the rational version, $e_*$ remains injective after tensoring with $\mathbb Q$.

The strong Novikov conjecture predicts when higher indices are nonzero, and the paper explains that the rational strong Novikov conjecture implies the Novikov conjecture. Its general validity is presented as an open problem.