Hairy Graphs Research Articles

Twisting of properads

We study T. Willwacher's twisting endofunctor tw in the category of dg prop(erad)s P under the operad of (strongly homotopy) Lie algebras, i:Lie→P. It is proven that if P is a properad under properad of Lie bialgebras Lieb, then the associated twisted properad twP becomes in general a properad under quasi-Lie bialgebras (rather than under Lieb). This result implies that the cyclic cohomology of any cyclic homotopy associative algebra has in general an induced structure of a quasi-Lie bialgebra. We show that the cohomology of the twisted properad twLieb is highly non-trivial — it contains the cohomology of the so called hairy graph complex introduced and studied recently in the context of the theory of long knots and the theory of moduli spaces Mg,n of algebraic curves of arbitrary genus g with n punctures.Using a polydifferential functor from the category of props to the category of operads, we introduce and study two new twisting endofunctors, one in the category dg prop(erad)s P under the minimal resolution of Lieb, and one for the involutive version of Lieb. We compute the cohomology of the associated deformation complexes, and discuss their applications in string topology.

Pre-Lie Pairs and Triviality of the Lie Bracket on the Twisted Hairy Graph Complexes

Abstract We study pre-Lie pairs, by which we mean a pair of a homotopy Lie algebra and a pre-Lie algebra with a compatible pre-Lie action. Such pairs provide a wealth of algebraic structure, which in particular can be used to analyze the homotopy Lie part of the pair. Our main application and the main motivation for this development are the dg Lie algebras of hairy graphs computing the rational homotopy groups of the mapping spaces of the little disks operads. We show that twisting with certain Maurer–Cartan elements trivializes their Lie algebra structure. The result can be used to understand the rational homotopy type of many connected components of these mapping spaces.

Multi-directed Graph Complexes and Quasi-isomorphisms Between Them II: Sourced Graphs

Abstract We prove that the projection from graph complex with at least one source to oriented graph complex is a quasi-isomorphism, showing that homology of the “sourced” graph complex is also equal to the homology of standard Kontsevich’s graph complex. This result may have applications in theory of multi-vector fields $T_{\textrm{poly}}^{\geq 1}$ of degree at least one, and to the hairy graph complex that computes the rational homotopy of the space of long knots. The result is generalized to multi-directed graph complexes, showing that all such graph complexes are quasi-isomorphic. These complexes play a key role in the deformation theory of multi-oriented props recently invented by Sergei Merkulov. We also develop a theory of graph complexes with arbitrary edge types.

Differentials on graph complexes III: hairy graphs and deleting a vertex

We continue studying the cohomology of the hairy graph complexes which compute the rational homotopy of embedding spaces, generalizing the Vassiliev invariants of knot theory, after the second part in this series. In that part we have proven that the hairy graph complex $$\mathrm {HGC}_{m,n}$$ with the extra differential is almost acyclic for even m. In this paper, we give the expected same result for odd m. As in the previous part, our results yield a way to construct many hairy graph cohomology classes by the waterfall mechanism also for odd m. However, the techniques are quite different. The main tool used in this paper is a new differential, deleting a vertex in non-hairy Kontsevich’s graphs, and a similar map for hairy vertices. We hope that the new differential can have further applications in the study of Kontsevich’s graph cohomology. Namely it is conjectured that the Kontsevich’s graph complex with deleting a vertex as an extra differential is acyclic.

Differentials on graph complexes II: hairy graphs

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 nonzero hairy graph cohomology classes out of (known) non-hairy classes by studying the cancellations in those sequences. This provide a first glimpse at the tentative global structure of the hairy graph cohomology.

Commutative hairy graphs and representations of Out (Fr)

We express the hairy graph complexes computing the rational homotopy groups of long embeddings (modulo immersion) of R^m in R^n as "decorated" graph complexes associated to certain representations of the outer automorphism groups of free groups. This interpretation gives rise to a natural spectral sequence, which allows us to shed some light on the structure of the hairy graph cohomology. We also explain briefly the connection to the deformation theory of the little discs operads and some conclusions that this brings.

Hopf algebras and invariants of the Johnson cokernel

We show that if H is a cocommutative Hopf algebra, then there is a natural action of Aut(F_n) on the nth tensor power of H which induces an Out(F_n) action on a quotient \overline{H^{\otimes n}}. In the case when H=T(V) is the tensor algebra, we show that the invariant Tr^C of the cokernel of the Johnson homomorphism studied in [J. Conant, The Johnson cokernel and the Enomoto-Satoh invariant, Algebraic and Geometric Topology, 15 (2015), no. 2, 801--821.] projects to take values in the top dimensional cohomology of Out(F_n) with coefficients in \overline{H^{\otimes n}}. We analyze the n=2 case, getting large families of obstructions generalizing the abelianization obstructions of [J. Conant, M. Kassabov, K. Vogtmann, Higher hairy graph homology, Journal of Topology, Geom. Dedicata 176 (2015), 345--374.].

Higher hairy graph homology

We study the hairy graph homology of a cyclic operad; in particular we show how to assemble corresponding hairy graph cohomology classes to form cocycles for ordinary graph homology, as defined by Kontsevich. We identify the part of hairy graph homology coming from graphs with cyclic fundamental group as the dihedral homology of a related associative algebra with involution. For the operads $$\mathsf{Comm}$$ $$\mathsf{Assoc}$$ and $$\mathsf{Lie}$$ we compute this algebra explicitly, enabling us to apply known results on dihedral homology to the computation of hairy graph homology. In addition we determine the image in hairy graph homology of the trace map defined in Conant et al. (J Topology 6(1):119–153, 2013), as a symplectic representation. For the operad $$\mathsf{Lie}$$ assembling hairy graph cohomology classes yields all known non-trivial rational homology of $$Out(F_n)$$ . The hairy graph homology of $$\mathsf{Lie}$$ is also useful for constructing elements of the cokernel of the Johnson homomorphism of a once-punctured surface.

Hairy graphs and the unstable homology of Mod(g , s ), Out(F n ) and Aut(F n )

We study a family of Lie algebras hO which are defined for cyclic operads O. Using his graph homology theory, Kontsevich identified the homology of two of these Lie algebras (corresponding to the Lie and associative operads) with the cohomology of outer automorphism groups of free groups and mapping class groups of punctured surfaces, respectively. In this paper, we introduce a hairy graph homology theory for O. We show that the homology of hO embeds in hairy graph homology via a trace map that generalizes the trace map defined by Morita. For the Lie operad, we use the trace map to find large new summands of the abelianization of hO which are related to classical modular forms for SL2(ℤ). Using cusp forms, we construct new cycles for the unstable homology of Out(Fn), and using Eisenstein series, we find new cycles for Aut(Fn). For the associative operad, we compute the first homology of the hairy graph complex by adapting an argument of Morita, Sakasai and Suzuki, who determined the complete abelianization of hO in the associative case.