On the delooping of (framed) embedding spaces

Ducoulombier, Julien; Turchin, Victor; Willwacher, Thomas

doi:10.1090/tran/8451

On the delooping of (framed) embedding spaces
HTML articles powered by AMS MathViewer

by Julien Ducoulombier, Victor Turchin and Thomas Willwacher PDF

Trans. Amer. Math. Soc. 374 (2021), 7657-7677 Request permission

Abstract:

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.

References

Michael Batanin and Florian De Leger, Polynomial monads and delooping of mapping spaces, J. Noncommut. Geom. 13 (2019), no. 4, 1521–1576. MR 4059827, DOI 10.4171/jncg/355
Clemens Berger and Ieke Moerdijk, The Boardman-Vogt resolution of operads in monoidal model categories, Topology 45 (2006), no. 5, 807–849. MR 2248514, DOI 10.1016/j.top.2006.05.001
Pedro Boavida de Brito and Michael Weiss, Spaces of smooth embeddings and configuration categories, J. Topol. 11 (2018), no. 1, 65–143. MR 3784227, DOI 10.1112/topo.12048
Ryan Budney, Little cubes and long knots, Topology 46 (2007), no. 1, 1–27. MR 2288724, DOI 10.1016/j.top.2006.09.001
Ryan Budney, An operad for splicing, J. Topol. 5 (2012), no. 4, 945–976. MR 3001316, DOI 10.1112/jtopol/jts024
Dan Burghelea and Richard Lashof, The homotopy type of the space of diffeomorphisms. I, II, Trans. Amer. Math. Soc. 196 (1974), 1–36; ibid. 196 (1974), 37–50. MR 356103, DOI 10.1090/S0002-9947-1974-0356103-2
Julien Ducoulombier, Delooping derived mapping spaces of bimodules over an operad, J. Homotopy Relat. Struct. 14 (2019), no. 2, 411–453. MR 3947966, DOI 10.1007/s40062-018-0217-3
Julien Ducoulombier, Delooping of high-dimensional spaces of string links, arXiv:1809.00682 (2018).
Julien Ducoulombier, Projective and Reedy model category structures for (infinitesimal) bimodules over an operad, arXiv:1911.03890 (2019).
Julien Ducoulombier, Delooping the functor calculus tower, arXiv:1708.02203 (2017)
William Dwyer and Kathryn Hess, Long knots and maps between operads, Geom. Topol. 16 (2012), no. 2, 919–955. MR 2928985, DOI 10.2140/gt.2012.16.919
Benoit Fresse, Homotopy of operads and Grothendieck-Teichmüller groups. Part 2, Mathematical Surveys and Monographs, vol. 217, American Mathematical Society, Providence, RI, 2017. The applications of (rational) homotopy theory methods. MR 3616816, DOI 10.1090/surv/217.2
Syunji Moriya and Keiichi Sakai, The space of short ropes and the classifying space of the space of long knots, Algebr. Geom. Topol. 18 (2018), no. 5, 2859–2873. MR 3848401, DOI 10.2140/agt.2018.18.2859
Jacob Mostovoy, Short ropes and long knots, Topology 41 (2002), no. 3, 435–450. MR 1910036, DOI 10.1016/S0040-9383(00)00047-1
Keiichi Sakai, Deloopings of the spaces of long embeddings, Fund. Math. 227 (2014), no. 1, 27–34. MR 3247031, DOI 10.4064/fm227-1-3
Paolo Salvatore, Knots, operads, and double loop spaces, Int. Math. Res. Not. , posted on (2006), Art. ID 13628, 22. MR 2276349, DOI 10.1155/IMRN/2006/13628
Victor Turchin, Delooping totalization of a multiplicative operad, J. Homotopy Relat. Struct. 9 (2014), no. 2, 349–418. MR 3258687, DOI 10.1007/s40062-013-0032-9
Alexander A. Voronov, The Swiss-cheese operad, Homotopy invariant algebraic structures (Baltimore, MD, 1998) Contemp. Math., vol. 239, Amer. Math. Soc., Providence, RI, 1999, pp. 365–373. MR 1718089, DOI 10.1090/conm/239/03610