On truncated mapping spaces of E_n operads

2019/11/02 by Thomas Willwacher, Willwacher, Thomas
Mathematics · Medicine · #Advanced Topics in Algebra #Algebraic Topology (math.AT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Ophthalmology and Eye Disorders #Quantum Algebra (math.QA)

paper · pdf · doi:10.48550/arxiv.1911.00709

openalex publication_date 2019/11/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The mapping spaces of the r-truncated versions of the E_n operads appear as the r-th stage of the Taylor tower for long embedding spaces. It has been shown that their rational homotopy groups can be expressed through graph homology in the limit r→ ∞. For finite r only a part of the graph homology appears in the homotopy groups of the Taylor tower at this stage, possibly along with some additional unstable homotopy groups. In this paper we study the convergence properties in some more detail. In particular, we provide bounds for the stage r at which the various graph homology classes start appearing, and we provide degree bounds for the unstable (i.e., vanishing as r→ ∞) terms in the homotopy groups of the tower.

Related

• The rational homotopy of mapping spaces of E_n operads
• Intrinsic convergence of the homological Taylor tower for r-immersions in \mathbb Rⁿ
• The homotopy groups of the inverse limit of a tower of fibrations
• Graph-complexes computing the rational homotopy of high dimensional analogues of spaces of long knots
• ∞-operadic foundations for embedding calculus