The rational homotopy of mapping spaces of E${}_n$ operads

TL;DR

This paper characterizes the rational homotopy type of mapping spaces between little discs operads using graph complexes, enabling computations of homotopy groups and insights into embeddings and operad structures.

Contribution

It expresses the rational homotopy type of these mapping spaces in terms of graph complexes and computes low-degree homotopy groups, revealing new non-trivial classes and rational equivalences.

Findings

Computed rational homotopy groups in low degrees.

Constructed infinite series of non-trivial homotopy classes.

Established rational equivalences for certain mapping spaces.

Abstract

We express the rational homotopy type of the mapping spaces $\mathrm{Map}^h(\mathsf D_m,\mathsf D_n^{\mathbb Q})$ of the little discs operads in terms of graph complexes. Using known facts about the graph homology this allows us to compute the rational homotopy groups in low degrees, and construct infinite series of non-trivial homotopy classes in higher degrees. Furthermore we show that for $n-m>2$, the spaces $\mathrm{Map}^h(\mathsf D_m,\mathsf D_n^{\mathbb Q})$ and $\mathrm{Map}^h(\mathsf D_m,\mathsf D_n)$ are simply connected and rationally equivalent. As application we determine the rational homotopy type of the deloopings of spaces of long embeddings. Some of the results hold also for mapping spaces $\mathrm{Map}_{\leq k}^h(\mathsf D_m,\mathsf D_n^{\mathbb Q})$, $\mathrm{Map}_{\leq k}^h(\mathsf D_m,\mathsf D_n)$, $n-m\geq 2$, of the truncated little discs operads, which allows one to determine rationally the delooping of the Goodwillie-Weiss tower for the spaces of long embeddings.