Brunnian links and Kontsevich graph complex I

We construct a natural chain map from the Kontsevich graph complex to the rational singular chain complex of $B{\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}_{\mathrm{\partial }}(D^{2k})$ 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 $B{\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}_{\mathrm{\partial }}(D^{2k})$ which are determined by well-known cycles in the graph complex.