Brunnian links and Kontsevich graph complex I

TL;DR

This paper constructs a topological realization of the Kontsevich graph complex by mapping it to the rational singular chain complex of diffeomorphism groups of disks, extending surgery theories to higher dimensions.

Contribution

It introduces a natural chain map from the Kontsevich graph complex to the rational chains of diffeomorphism groups, generalizing 3-manifold surgery theories to higher dimensions.

Findings

Established a chain map linking graph complex to diffeomorphism groups

Provided new elements in the rational homotopy groups of diffeomorphism groups

Generalized surgery theories from 3-manifolds to higher-dimensional disks

Abstract

We construct a natural chain map from the Kontsevich graph complex to the rational singular chain complex of $B\mathrm{Diff}_\partial(D^{2k})$ when the dimension $2k$ is sufficiently large, generalizing Goussarov and Habiro's theories of surgery on 3-valent graphs in 3-manifolds. Our construction can be considered as a topological realization of the Kontsevich graph complex. We also give new constructions of elements in the rational homotopy groups of $B\mathrm{Diff}_\partial(D^{2k})$ which are determined by well-known cycles in the graph complex.