  Linking numbers in higher dimensions and their generalization including gauge
fields are studied in the context of BF theories. The linking numbers
associated to $n$-manifolds with smooth flows generated by divergence-free
p-vector fields, endowed with an invariant flow measure are computed in
different cases. They constitute invariants of smooth dynamical systems (for
non-singular flows) and generalizes previous results for the 3-dimensional
case. In particular, they generalizes to higher dimensions the Arnold's
asymptotic Hopf invariant for the three-dimensional case. This invariant is
generalized by a twisting with a non-abelian gauge connection. The computation
of the asymptotic Jones-Witten invariants for flows is naturally extended to
dimension n=2p+1. Finally we give a possible interpretation and implementation
of these issues in the context of string theory.
