?
Differentials on graph complexes
Advances in Mathematics. 2017. Vol. 307. P. 1184–1214.
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
Merkulov S., Willwacher T., Letters in Mathematical Physics 2014 Vol. 104 No. 5 P. 625–634
Added: October 1, 2026
Merkulov S., American Mathematical Society, 2021.
Introduction into the theory of the Grothendieck-Teichmueller group ...
Added: September 30, 2026
Merkulov S., Letters in Mathematical Physics 2020 Vol. 110 P. 1425–1475
We introduce a new category of differential graded {\em multi-oriented}\, props whose representations (called homotopy algebras with branes) in a graded vector space require a choice of a collection of $k$ linear subspaces in that space, $k$ being the number of extra directions (if $k=0$ this structure recovers an ordinary prop); symplectic vector spaces equipped ...
Added: September 30, 2026
Merkulov S., Journal of Pure and Applied Algebra 2026 Vol. 230 P. 1–19
We study the dual cyclic Hochschild complex $Cyc(A,\K)$ of a
(possibly, infinite-dimensional) $A_\infty$-algebra $(A,\mu)$ and prove
that any pre-Calabi-Yau extension $\pi$ of the given $A_\infty$ structure $\mu$ in $A$
induces on the cyclic cohomology of $(A,\mu)$ a representation of a new dg properad of {\em oriented}\, ribbon graphs. We compute the cohomology of that properad in terms of ...
Added: September 29, 2026
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
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
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
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
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
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
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