Higher-order jet and its application to the Novikov conjecture

In this paper, we apply higher-order tangent bundles and higher-order derivative maps to study countable subgroups of the deffeomorphism group ${\rm Diff}(M)$ of a closed smooth manifold $M$. Using this framework, we introduce the concept of higher order $μ$-discreteness for subgroup $Γ< {\rm Diff}(M)$. As applications, we show that both the strong Novikov conjecture and the coarse Baum--Connes conjecture hold for countable groups with higher order $μ$-discreteness.