An Adiabatic Decomposition of the Hodge Cohomology of Manifolds Fibred over Graphs

TL;DR

This paper develops an adiabatic decomposition of Hodge cohomology for manifolds fibred over graphs, combining topological and analytical methods, extending previous work to multiple edges and focusing on the Gauss-Bonnet operator.

Contribution

It introduces a new adiabatic decomposition framework for manifolds with embedded cylinders over graphs, generalizing prior single-edge results and incorporating the Cappell-Lee-Miller splicing map.

Findings

Describes the adiabatic behavior of harmonic forms using a cech-de Rham complex

Generalizes the splicing map to multiple edges

Focuses on the Gauss-Bonnet operator in the decomposition

Abstract

In this article we use the combinatorial and geometric structure of manifolds with embedded cylinders in order to develop an adiabatic decomposition of the Hodge cohomology of these manifolds. We will on the one hand describe the adiabatic behaviour of spaces of harmonic forms by means of a certain \v{C}ech-de Rham complex and on the other hand generalise the Cappell-Lee-Miller splicing map to the case of a finite number of edges, thus combining the topological and the analytic viewpoint. In parts, this work is a generalisation of works of Cappell, Lee and Miller in which a single-edged graph is considered, but it is more specific since only the Gauss-Bonnet operator is studied.