Differentials on graph complexes

A. Khoroshkin; Willwacherb T.; Živković M.

doi:10.1016/j.aim.2016.05.029

Publications

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Differentials on graph complexes

Advances in Mathematics. 2017. Vol. 307. P. 1184–1214.

Khoroshkin A., Willwacherb T., Živković M.

We study the cohomology of complexes of ordinary (non-decorated) graphs, introduced by M. Kontsevich. We construct spectral sequences converging to zero whose first page contains the graph cohomology. In particular, these spectral sequences may be used to show the existence of an infinite series of previously unknown and provably non-trivial cohomology classes, and put constraints on the structure of the graph cohomology as a whole.

Keywords: graph complexes

Quantizations of Lie bialgebras, duality involution and oriented graph complexes

Merkulov S., Živković M., Letters in Mathematical Physics 2022 Vol. 112 No. 13

We prove that the action of the Grothendieck–Teichmüller group on the genus completed properad of (homotopy) Lie bialgebras commutes with the reversing directions involution of the latter. We also prove that every universal quantization of Lie bialgebras is homotopy equivalent to the one which commutes with the duality involution exchanging Lie bracket and Lie cobracket. ...

Added: December 19, 2025

The oriented graph complex revisited

Merkulov S., Willwacher T., Wolff V., Letters in Mathematical Physics 2025 Vol. 115 Article 117

We prove that the Kontsevich graph complex \GC_d^{2}$ and its oriented version $OGC_{d+1}^2$ are quasi-isomorphic as dg Lie algebras. ...

Added: November 1, 2025

On interrelations between graph complexes

Sergei A. Merkulov, International Mathematics Research Notices 2025 Vol. 2025 No. 2 Article rnae287

We study Maxim Kontsevich's graph complex $\GCd$ for any integer $d$ as well as its oriented and targeted versions, and show new short proofs of the theorems due to Thomas Willwacher and Marko \v Zivkovi\' c which establish isomorphisms of their cohomology groups. A new result relating the cohomology of the sourced-targeted graph complex in ...

Added: October 10, 2025

Supersymmetry and cohomology of graph complexes.

Barannikov S., Letters in Mathematical Physics 2019 Vol. 109 No. 3 P. 699–724

https://arxiv.org/abs/1803.11549 I describe a combinatorial construction of the cohomology classes in compactified moduli spaces of curves ZˆI∈H∗(barM_g,n) starting from the following data: an odd derivation I, whose square is non-zero in general, I2≠0, acting on a ℤ/2ℤ-graded associative algebra with odd scalar product. The constructed cocycles were first described in the theorem 2 in the author's paper "Noncommmutative Batalin-Vilkovisky geometry and ...

Added: October 5, 2018

Topology of complexes of edge covering partite graphs and hypergraphs

Victor A. Vassiliev, Combinatorica 2018 Vol. 38 No. 5 P. 1239–1249

We describe the homotopy types of complexes of partite graphs and hypergraphs with a fixed set of vertices covered by their edges ...

Added: December 27, 2017

Differentials on Graph Complexes II, Hairy graphs

Willwacher T., Zivkovic M., / Cornell University. Серия "Working papers by Cornell University". 2015. № 1508.01281.

We study the cohomology of the hairy graph complexes which compute the rational homotopy of embedding spaces, generalizing the Vassiliev invariants of knot theory. We provide spectral sequences converging to zero whose first pages contain the hairy graph cohomology. Our results yield a way to construct many hairy graph cohomology classes out of non-hairy classes ...

Added: December 14, 2015

Differentials on graph complexes

Khoroshkin A., Willwacher T., Živković M., / Series math "arxiv.org". 2014. No. 1411.2369.

We study the cohomology of complexes of ordinary (non-decorated) graphs, introduced by M. Kontsevich. We construct spectral sequences converging to zero whose first page contains the graph cohomology. In particular, these series may be used to show the existence of an infinite series of previously unknown and provably non-trivial cohomology classes, and put constraints on ...

Added: December 9, 2014