The rational homotopy of mapping spaces of E_n operads

2017/03/17 by Benoît Fresse, Fresse, Benoit, Victor Turchin +3 · 3 citations
Mathematics · #Advanced Topics in Algebra #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA)

paper · pdf · doi:10.48550/arxiv.1703.06123

openalex publication_date 2017/03/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We express the rational homotopy type of the mapping spaces 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 Map^h(\mathsf D_m,\mathsf D_n^{\mathbb Q}) and 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 Map_{≤ k}^h(\mathsf D_m,\mathsf D_n^{\mathbb Q}), Map_{≤ k}^h(\mathsf D_m,\mathsf D_n), n-m≥ 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.

Citations

• Relative (non-)formality of the little cubes operads and the algebraic Cerf Lemma
• Operads, homotopy algebra and iterated integrals for double loop spaces

Cited by

• The 11-loop graph cohomology
• On the homotopy type of the spaces of spherical knots in Rⁿ
• Oriented hairy graphs and moduli spaces of curves

Related

• On truncated mapping spaces of E_n operads
• Mapping Spaces for DG Hopf Cooperads and Homotopy Automorphisms of the Rationalization of E_n-operads
• Little discs operads, graph complexes and Grothendieck--Teichmüller groups
• Graph-complexes computing the rational homotopy of high dimensional analogues of spaces of long knots
• Two-loop part of the rational homotopy of spaces of long embeddings