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

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 ${\mathrm{Map}}_{\le k}^h({\mathsf{D}}_m,{\mathsf{D}}_n^{\mathbb{Q}})$, ${\mathrm{Map}}_{\le k}^h({\mathsf{D}}_m,{\mathsf{D}}_n)$, $n-m\ge 2$, 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.

References in corpus (4)

• Operads, homotopy algebra and iterated integrals for double loop spaces
• Embeddings from the point of view of immersion theory: Part I
• Long knots and maps between operads
• Relative (non-)formality of the little cubes operads and the algebraic Cerf Lemma

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