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

Arone, Gregory; Turchin, Victor

Annales de l'Institut Fourier, Tome 65 (2015) no. 1, pp. 1-62.

On continue notre étude des espaces de plongements longs (les plongements longs sont des analogues en dimension supérieure des nœuds longs). Dans notre travail précédent, on a montré que dans le cas où les dimensions sont dans le rang stable l’homologie rationnelle de ces espaces peut être calculée comme l’homologie d’un certain complexe de graphes que l’on a décrit explicitement. Dans ce travail, on établit un résultat similaire pour les groupes d’homotopie rationnelle de ces espaces. On met aussi un accent sur les différentes façons d’effectuer ces calculs. En particulier, on décrit trois complexes de graphes différents calculant les groupes d’homotopie en question. On calcule également les fonctions génératrices des caractéristiques eulériennes des termes d’une décomposition en somme directe des complexes calculant les groupes d’homologie.

We continue our investigation of spaces of long embeddings (long embeddings are high-dimensional analogues of long knots). In previous work we showed that when the dimensions are in the stable range, the rational homology groups of these spaces can be calculated as the homology of a direct sum of certain finite graph-complexes, which we described explicitly. In this paper, we establish a similar result for the rational homotopy groups of these spaces. We also put emphasis on the different ways the calculations can be done. In particular we describe three different graph-complexes computing these rational homotopy groups. We also compute the generating functions of the Euler characteristics of the summands in the homological splitting.

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