Double graph complex and characteristic classes of fibrations

TL;DR

This paper develops a double graph complex and a chain map to study characteristic classes of fibrations, linking graph cohomology with derivations of symplectic structures to understand fiber invariants.

Contribution

It introduces a novel double chain complex of graphs and a chain map to the Chevalley-Eilenberg complex, connecting graph theory with characteristic classes of fibrations.

Findings

Characteristic classes of fibrations described via graph cohomology

Chain map relates graph complex to derivations of symplectic structures

Provides a new method to compute fiber invariants using graphs

Abstract

In this paper, we construct a double chain complex generated by certain graphs and a chain map from that to the Chevalley-Eilenberg double complex of the dgl of symplectic derivations on a free dgl. It is known that the target of the map is related to characteristic classes of fibrations. We can describe some characteristic classes of fibrations whose fiber is a 1-punctured even-dimensional manifold by linear combinations of graphs though the cohomology of the dgl of derivations.