Commutative hairy graphs and representations of Out (Fr)

Abstract

We express the hairy graph complexes computing the rational homotopy groups of long embeddings (modulo immersions) 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 disks operads and some conclusions that this brings. . † For a weaker range of dimensions this result has been shown earlier in [2,21]. To recall, the space Embc(R m , R n ) of long embeddings modulo immersions is the homotopy fiber of the inclusion Embc(R m , R n ) → Immc(R m , R n ) of the space of long embeddings R m → R n (that is, embeddings coinciding with the fixed linear inclusion R m ⊂ R n outside a compact subset of R m ) into the space of long immersions (that is, immersions with the same behavior at infinity). The rational homotopy groups of Embc(R m , R n ) and those of Embc(R m , R n ) are the same up to a finite dimensional correction, which was computed in [2].

Homology representations of compactified configurations on graphs applied to $\mathcal{M}_{2,n}$

“…It generalizes the ordinary Hochschild homology when specializing to the homology of S 1 with suitable coefficients, see [Pir00]. Specializing instead to the space R g , the resulting homology with its Out(F g )-action is related to homology of hairy-graph complexes, and to rational homotopy of spaces of long embeddings R m ֒→ R n for n − m ≥ 3, see [TW17].…”

Hairy graphs to ribbon graphs via a fixed source graph complex

“…Again, because the group is finite, the space of invariants may be replaced by the space of coinvariants. The hairy graph complex with s hairs is (16) H s GC n := v≥1,e≥0…”

“…Theorem 1.6 ( [15], [16], [17], [6]). There is a quasi-isomorphism GC 2 n , d → HGC 2 n,n , d + h , by summing over all ways to attach a hair to each graph.…”

Hochschild-Pirashvili homology on suspensions and representations of Out(F_n)

“…To finish the proof we notice that the composite of the first arrow in (34) encoding the rational homotopy type of f : ΣY * → ΣZ * , determines the map of higher Hochschild complexes CH (−) (−) (in fact we will work with gr CH (−) (−) instead). For simplicity we will be assuming that Y * and Z * are of finite type and we will only look at the case L * = M ⊗ C(g) ⊗• , where g is strictly positively graded.…”

“…acknowledges partial support by the Swiss National Science Foundation (grant 200021 150012) and the SwissMap NCCR, funded by the Swiss National Science Foundation. 1 These representations appear as application to the hairy graph-homology computations in the study of the spaces of long embeddings, higher dimensional string links, and the deformation theory of the little discs operads [2,31,32,33,34].…”