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Register a new userOn the delooping of (framed) embedding spaces
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It is known that the bimodule derived mapping spaces between two operads have a delooping in terms of the operadic mapping space. We show a relative version of that statement. The result has applications to the spaces of disc embeddings fixed near the boundary and framed disc embeddings.
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We extend some classical results - such as Quillens Theorem A, the Grothendieck construction, Thomasons Theorem and the characterisation of homotopically cofinal functors - from the homotopy theory of small categories to polynomial monads and their algebras. As an application we give a categorical proof of the Dwyer-Hess and Turchin results concerning the explicit double delooping of spaces of long knots.
From a map of operads $eta : Orightarrow O$, we introduce a cofibrant replacement of the operad $O$ in the category of bimodules over itself such that the corresponding model of the derived mapping space of bimodules $Bimod_{O}^{h}(O;O)$ is an algebra over the one dimensional little cubes operad $mathcal{C}_{1}$. In the present work, we also build an explicit weak equivalence of $mathcal{C}_{1}$-algebras from the loop space $Omega Operad^{h}(O;O)$ to $Bimod_{O}^{h}(O;O)$.
We study configuration spaces of framed points on compact manifolds. Such configuration spaces admit natural actions of the framed little discs operads, that play an important role in the study of embedding spaces of manifolds and in factorization homology. We construct real combinatorial models for these operadic modules, for compact smooth manifolds without boundary.
We study a connection between mapping spaces of bimodules and of infinitesimal bimodules over an operad. As main application and motivation of our work, we produce an explicit delooping of the manifold calculus tower associated to the space of smooth maps $D^{m}rightarrow D^{n}$ of discs, $ngeq m$, avoiding any given multisingularity and coinciding with the standard inclusion near $partial D^{m}$. In particular, we give a new proof of the delooping of the space of disc embeddings in terms of little discs operads maps with the advantage that it can be applied to more general mapping spaces.
In the world of chain complexes E_n-algebras are the analogues of based n-fold loop spaces in the category of topological spaces. Fresse showed that operadic E_n-homology of an E_n-algebra computes the homology of an n-fold algebraic delooping. The aim of this paper is to construct two spectral sequences for calculating these homology groups and to treat some concrete classes of examples such as Hochschild cochains, graded polynomial algebras and chains on iterated loop spaces. In characteristic zero we gain an identification of the summands in Pirashvilis Hodge decomposition of higher order Hochschild homology in terms of derived functors of indecomposables of Gerstenhaber algebras and as the homology of exterior and symmetric powers of derived Kahler differentials.
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