Goodwillie's cosimplicial model for the space of long knots and its applications

Abstract

We work out the details of a correspondence observed by Goodwillie between cosimplicial spaces and good functors from a category of open subsets of the interval to the category of compactly generated weak Hausdorff spaces. Using this, we compute the first page of the integral Bousfield--Kan homotopy spectral sequence of the tower of fibrations given by the Taylor tower of the embedding functor associated to the space of long knots. Based on the methods in [Con08], we give a combinatorial interpretation of the differentials $d^1$ mapping into the diagonal terms, by introducing the notion of $(i,n)$-marked unitrivalent graphs.

arXiv commentRevised according to referee report. Published in the Journal of Homotopy and Related Structures

References in corpus (10)

• Higher Topos Theory
• Claspers and finite type invariants of links
• Embeddings from the point of view of immersion theory: Part I
• Factorization homology of topological manifolds
• Embeddings from the point of view of immersion theory: Part II
• Long knots and maps between operads
• A family of embedding spaces
• Covariant Model Structures and Simplicial Localization
• Embedding calculus knot invariants are of finite type
• Embedding calculus and grope cobordism of knots