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

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 different ways how the calculations can be done. In particular we describe three different graph-complexes computing the rational homotopy of spaces of long embeddings. We also compute the generating functions of the Euler characteristics of the summands in the homological splitting.

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