287 research outputs found

    From maps between coloured operads to Swiss-Cheese algebras

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    In the present work, we extract pairs of topological spaces from maps between coloured operads. We prove that those pairs are weakly equivalent to explicit algebras over the one dimensional Swiss-Cheese operad SC_{1}. Thereafter, we show that the pair formed by the space of long knots and the polynomial approximation of (k)-immerions from R^{d} to R^{n} is an SC_{d+1}-algebra assuming the Dwyer-Hess'conjecture

    On Operadic Actions on Spaces of Knots and 2-Links

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    In the present work, we realize the space of string 2-links L\mathcal{L} as a free algebra over a colored operad denoted SCL\mathcal{SCL} (for "Swiss-Cheese for links"). This result extends works of Burke and Koytcheff about the quotient of L\mathcal{L} by its center and is compatible with Budney's freeness theorem for long knots. From an algebraic point of view, our main result refines Blaire, Burke and Koytcheff's theorem on the monoid of isotopy classes of string links. Topologically, it expresses the homotopy type of the isotopy class of a string 2-link in terms of the homotopy types of the classes of its prime factors.Comment: Comments are welcom

    On 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.Comment: arXiv admin note: text overlap with arXiv:1704.0706

    A model for configuration spaces of points

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    The configuration space of points on a DD-dimensional smooth framed manifold may be compactified so as to admit a right action over the framed little DD-disks operad. We construct a real combinatorial model for these modules, for compact smooth manifolds without boundary

    Delooping the functor calculus tower

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    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 DmDnD^{m}\rightarrow D^{n} of discs, nmn\geq m, avoiding any given multisingularity and coinciding with the standard inclusion near Dm\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

    Boardman-Vogt resolutions and bar/cobar constructions of (co)operadic (co)bimodules

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    We develop the combinatorics of leveled trees in order to construct explicit resolutions of (co)operads and (co)operadic (co)bimodules. We build explicit cofibrant resolutions of operads and operadic bimodules in spectra analogous to the ordinary Boardman--Vogt resolutions and we express them as cobar constructions of indecomposable elements. Dually, in the context of CDGAs, we perform similar constructions, and we obtain fibrant resolutions of Hopf cooperads and Hopf cooperadic cobimodules. We also express them as bar constructions of primitive elements

    Projective and Reedy model category structures for (infinitesimal) bimodules over an operad

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    We construct and study projective and Reedy model category structures for bimodules and infinitesimal bimodules over topological operads. Both model structures produce the same homotopy categories. For the model categories in question, we build explicit cofibrant and fibrant replacements. We show that these categories are right proper and under some conditions left proper. We also study the extension/restriction adjunctions.Comment: All comments on this work are welcom
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