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

TL;DR

This paper extends the understanding of high-dimensional long embedding spaces by establishing graph-complexes that compute their rational homotopy groups, complementing previous homology results and providing multiple computational approaches.

Contribution

It introduces three different graph-complexes for calculating the rational homotopy groups of long embedding spaces in high dimensions, expanding the computational tools available.

Findings

Established graph-complexes for rational homotopy computation

Computed generating functions for Euler characteristics

Provided multiple methods for calculations

Abstract

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.