Graph-complexes computing the rational homotopy of high dimensional analogues of spaces of long knots

Two-loop part of the rational homotopy of spaces of long embeddings

“…In this section we recall definition of the complex E m,N π of uni-≥3-valent graphs from [2]. The homology of this complex is naturally isomorphic to π * Emb c (R m , R N ) ⊗ Q, N ≥ 2m + 2.…”

“…For the part H m,N π (t + 1, t) ֒→ E m,N π (t + 1, t), i.e. the part which is spanned by trees, see [2,Proposition 3.2]. For the case when the first Betti number is positive the proof follows from the argument of [3,Theorem 1.1].…”

“…By Emb c (R m , R N ) we denote the homotopy fiber of the inclusion Emb c (R m , R N ) ֒→ Imm c (R m , R N ), where Imm c (R m , R N ) is the space of immersions with the same behavior at infinity. In [1,2] Arone and Turchin study the rational homology and homotopy of Emb c (R m , R N ) and Emb c (R m , R N ). In particular in [2] the authors define graph-complexes E m,N π computing the rational homotopy π * Emb c (R m , R N ) ⊗ Q, N ≥ 2m + 2.…”

“…In [1,2] Arone and Turchin study the rational homology and homotopy of Emb c (R m , R N ) and Emb c (R m , R N ). In particular in [2] the authors define graph-complexes E m,N π computing the rational homotopy π * Emb c (R m , R N ) ⊗ Q, N ≥ 2m + 2. The graph-complexes up to a regrading depend only on the parities of m and N .…”

“…Thus there are only 4 cases to consider. The connection between π * Emb c (R m , R N ) ⊗ Q and π * Emb c (R m , R N ) ⊗ Q, N ≥ 2m + 2, is established by [2,Corollary 4.3]. The graph-complexes E m,N π split into a direct sum of complexes by the number of univalent vertices (this number is called Hodge degree) and the number of loops (first Betti number) in the graphs.…”