Spaces of smooth embeddings and configuration categories

Abstract

In the homotopical study of spaces of smooth embeddings, the functor calculus method (Goodwillie-Klein-Weiss manifold calculus) has opened up important connections to operad theory. Using this and a few simplifying observations, we arrive at an operadic description of the obstructions to deforming smooth immersions into smooth embeddings. We give an application which in some respects improves on recent results of Arone-Turchin and Dwyer-Hess concerning high-dimensional variants of spaces of long knots.

arXiv comment84 pages. v2: minor errors in section 7 corrected, old section 8 improved and reorganized into three separate sections. v3: typos corrected

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Cited by in corpus (22)

• A model for configuration spaces of points
• Some finiteness results for groups of automorphisms of manifolds
• Koszul duality for operadic categories
• The cohomology of Torelli groups is algebraic
• Dalian notes on rational Pontryagin classes
• On the formality of the little disks operad in positive characteristic
• Embedding calculus and smooth structures
• Galois symmetries of knot spaces
• Operadic categories as a natural environment for Koszul duality
• Delooping of high-dimensional spaces of string links
• On the homotopy type of the spaces of spherical knots in $R^n$
• A spectral sequence for spaces of maps between operads
• On truncated mapping spaces of $E_n$ operads
• Embedding calculus for surfaces
• Occupants in simplicial complexes
• Embedding calculus and grope cobordism of knots
• An Application of the $h$-principle to Manifold Calculus
• Low stages of the Taylor tower for r-immersions
• The Disc-structure space
• An infinity operad of normalized cacti
• Culf maps and edgewise subdivision
• Addendum to: Some exotic nontrivial elements of the rational homotopy groups of $\mathrm{Diff}(S^4)$ (homological interpretation)