A Trace Formula for Foliated Flows

This book presents a new Lefschetz trace formula for a foliated flow on a compact foliated manifold with a foliation of codimension one. The leaves preserved by the flow and its closed orbits are assumed to be transversely simple. The formula equates two distributions on the real line: one is a renormalized trace of the flow’s action on two reduced leafwise cohomologies, defined via conormal and dual-conormal currents; the other consists of contributions from the preserved leaves, closed orbits, and a b-trace version of Connes’ Euler characteristic, defined using a transverse invariant measure on the complement of the preserved leaves. The usual Euler characteristic is undefined here due to non-compactness. The proofs and definitions combine tools from analysis and foliation theory, like small b-calculus, Witten’s deformation of differential complexes, heat invariants, and zeta functions of operators, alongside a description of foliations with this type of flow. The exposition is largely self-contained, with prerequisites and references given for further study. The trace formula solves a conjecture of C. Deninger, motivated by his program which links arithmetic zeta functions to foliated geometry. While further generalization is needed for arithmetic applications, the original and technically deep ideas presented here are valuable in themselves and offer a foundation for future work. The book will be of interest to researchers and graduate students in foliation theory, global analysis on manifolds, and arithmetic geometry.