Holonomy, Zeta Functions, and Cohomological Structures in Foliated Manifolds with Stratified Boundaries

TL;DR

The paper addresses the interplay between holonomy, Ihara zeta functions, and cohomology in foliated manifolds with stratified boundaries by introducing ratified $\mathcal{F}$-completions and $\Gamma$-sets. It develops a geometric interface construction $\mathcal{I}$, defines an Ihara zeta function $Z(\Gamma; u)$ on the associated $\Gamma$-set, and analyzes $\Delta$-actions and the twist map $\tau$ to study holonomy and cohomology in stratified spaces. A central conjecture posits a holonomy fixed-point–zeta duality, linking holonomy fixed points to zeros/poles of a geodesic zeta function via monodromy, with extensions to twisted cohomology. The framework yields insights for spectral graph theory, tiling in $\mathbb{R}^n$, and topological invariants in geometric topology and mathematical physics, offering a structured perspective on symmetries, holonomy, and cohomology in complex foliated geometries.

Abstract

This paper explores the interplay between holonomy, Ihara zeta functions, and cohomological structures within the framework of ratified F-completions of foliated manifolds. We develop a novel formalism for the Gamma-set, a topological multigraph that captures intersection points of foliations with stratified boundaries, and use it to define an Ihara zeta function that encodes the manifold's symmetries. By investigating the holonomy group of spinor fields in relation to the Delta-actions on the Gamma-set, we conjecture a duality between holonomy fixed points and the poles of the Ihara zeta function, extending to twisted cohomology classes. We further analyze how the twist map T impacts cohomology within stratified spaces, highlighting the role of vector bundles and the twisted cochain complex. Applications of our framework to spectral graph theory and tiling in R^n demonstrate the power of holonomy-zeta duality in geometrically rich foliated structures.