arXiv subjects

Assar Andersson

Publications and source records attributed to Assar Andersson.

4 recordsLinked to original sources

arXiv2026

Wheel Classes in Kontsevich Graph Complex and Merkulov's Low-Valence Conjecture

We show that the wheel classes in the Kontsevich graph complex $GC_d$ admit representatives supported on graphs with only $3$- and $4$-valent vertices. This verifies that Merkulov's low-valence conjecture holds for the wheel classes. More precisely, for every $m \ge 2$, we prove that the wheel graph $W_{2m+1}$ is homologous to an explicit linear combination of $2^{m-2}$ graphs, each having only $3$- and $4$-valent vertices.

math.QA↗

arXiv2020

Oriented hairy graphs and moduli spaces of curves

We discuss a graph complex formed by directed acyclic graphs with external legs. This complex comes in particular with a map to the ribbon graph complex computing the (compactly supported) cohomology of the moduli space of points $\mathcal M_{g,n}$, extending an earlier result of Merkulov-Willwacher. It is furthermore quasi-isomorphic to the hairy graph complex computing the weight 0 part of the compactly supported cohomology of $\mathcal{M}_{g,n}$ according to Chan-Galatius-Payne. Hence we can naturally connect the works Chan-Galatius-Payne and of Merkulov-Willwacher and the ribbon graph complex and obtain a fairly satisfying picture of how all the pieces and various graph complexes fit together, at least in weight zero.

math.QA↗

arXiv2020

Hairy graphs to ribbon graphs via a fixed source graph complex

We show that the hairy graph complex $(HGC_{n,n},d)$ appears as an associated graded complex of the oriented graph complex $(OGC_{n+1},d)$, subject to the filtration on the number of targets, or equivalently sources, called the fixed source graph complex. The fixed source graph complex $(OGC_1,d_0)$ maps into the ribbon graph complex $RGC$, which models the moduli space of Riemann surfaces with marked points. The full differential $d$ on the oriented graph complex $OGC_{n+1}$ corresponds to the deformed differential $d+h$ on the hairy graph complex $HGC_{n,n}$, where $h$ adds a hair. This deformed complex $(HGC_{n,n},d+h)$ is already known to be quasi-isomorphic to standard Kontsevich's graph complex $GC^2_n$. This gives a new connection between the standard and the oriented version of Kontsevich's graph complex.

math.QA↗

arXiv2019

From deformation theory of wheeled props to classification of Kontsevich formality maps

We study homotopy theory of the wheeled prop controlling Poisson structures on arbitrary formal graded finite-dimensional manifolds and prove, in particular, that Grothendieck-Teichmueller group acts on that wheeled prop faithfully and homotopy non-trivially. Next we apply this homotopy theory to the study of the deformation complex of an arbitrary Maxim Kontsevich formality map and compute the full cohomology group of that deformation complex in terms of the cohomology of a certain graph complex introduced earlier by Maxim Kontsevich in [K1] and studied by Thomas Willwacher in [W1].

math.QA↗