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

TL;DR

This paper develops a new formalism connecting holonomy, zeta functions, and cohomology in stratified foliated manifolds, revealing dualities and applications in spectral graph theory and tilings.

Contribution

It introduces the Gamma-set formalism and explores the duality between holonomy fixed points and zeta function poles, extending to twisted cohomology in stratified spaces.

Findings

Gamma-set formalism captures foliation intersections with stratified boundaries

Holonomy-zeta duality relates fixed points to zeta poles

Applications demonstrated in spectral graph theory and tilings

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.