## The cleavage operad and string topology of higher dimension

HTML articles powered by AMS MathViewer

- by Tarje Bargheer PDF
- Trans. Amer. Math. Soc.
**366**(2014), 4209-4241 Request permission

## Abstract:

For a manifold $N$ embedded inside euclidean space $\mathbb {R}^{n+1}$, we produce a coloured operad that acts on the space of maps from $N$ to $M$, where $M$ is a compact, oriented, smooth manifold. Our main example of interest is $N$, the unit sphere, and we indicate how this gives homological actions, generalizing the action of the spineless cacti operad and retrieving the Chas-Sullivan product by taking $N$ to be the unit circle in $\mathbb {R}^2$. We go on to show that for $S^n$, the unit sphere in $\mathbb {R}^{n+1}$, the operad constructed is a coloured $E_{n+1}$-operad. This $E_{n+1}$-structure is finally twisted by $SO(n+1)$ to homologically agree with actions of the operad of framed little $(n+1)$-disks.## References

- Tarje Bargheer,
*Cultivating operads in string topology*, 2008, Master’s Thesis, University of Copenhagen, www.math.ku.dk/$\sim$bargheer. - —,
*A colourful approach to string topology*, 2011, PhD Thesis, University of Copenhagen, www.math.ku.dk/$\sim$bargheer. - —,
*Umkehr maps patched via the compactified cleavage operad*, 2012, preprint. - Clemens Berger,
*Combinatorial models for real configuration spaces and $E_n$-operads*, Operads: Proceedings of Renaissance Conferences (Hartford, CT/Luminy, 1995) Contemp. Math., vol. 202, Amer. Math. Soc., Providence, RI, 1997, pp. 37–52. MR**1436916**, DOI 10.1090/conm/202/02582 - Morton Brown,
*A proof of the generalized Schoenflies theorem*, Bull. Amer. Math. Soc.**66**(1960), 74–76. MR**117695**, DOI 10.1090/S0002-9904-1960-10400-4 - Moira Chas and Dennis Sullivan,
*String topology*, 1999, arXiv.org:math/9911159. - Ralph L. Cohen and John D. S. Jones,
*A homotopy theoretic realization of string topology*, Math. Ann.**324**(2002), no. 4, 773–798. MR**1942249**, DOI 10.1007/s00208-002-0362-0 - Ralph L. Cohen and John R. Klein,
*Umkehr maps*, Homology Homotopy Appl.**11**(2009), no. 1, 17–33. MR**2475820** - Ralph L. Cohen and Alexander A. Voronov,
*Notes on string topology*, String topology and cyclic homology, Adv. Courses Math. CRM Barcelona, Birkhäuser, Basel, 2006, pp. 1–95. MR**2240287** - Dan Dugger,
*A primer on homotopy colimits*, http://math.uoregon.edu/$\sim$ddugger/hocolim.pdf, 2008. - Yves Félix and Jean-Claude Thomas,
*String topology on Gorenstein spaces*, Math. Ann.**345**(2009), no. 2, 417–452. MR**2529482**, DOI 10.1007/s00208-009-0361-5 - Kate Gruher and Paolo Salvatore,
*Generalized string topology operations*, Proc. Lond. Math. Soc. (3)**96**(2008), no. 1, 78–106. MR**2392316**, DOI 10.1112/plms/pdm030 - Philip S. Hirschhorn,
*Model categories and their localizations*, Mathematical Surveys and Monographs, vol. 99, American Mathematical Society, Providence, RI, 2003. MR**1944041**, DOI 10.1090/surv/099 - Po Hu,
*Higher string topology on general spaces*, Proc. London Math. Soc. (3)**93**(2006), no. 2, 515–544. MR**2251161**, DOI 10.1112/S0024611506015838 - Sadok Kallel and Paolo Salvatore,
*Rational maps and string topology*, Geom. Topol.**10**(2006), 1579–1606. MR**2284046**, DOI 10.2140/gt.2006.10.1579 - Ralph M. Kaufmann,
*On several varieties of cacti and their relations*, Algebr. Geom. Topol.**5**(2005), 237–300. MR**2135554**, DOI 10.2140/agt.2005.5.237 - Tom Leinster,
*Higher operads, higher categories*, London Mathematical Society Lecture Note Series, vol. 298, Cambridge University Press, Cambridge, 2004. MR**2094071**, DOI 10.1017/CBO9780511525896 - James E. McClure and Jeffrey H. Smith,
*Operads and cosimplicial objects: an introduction*, Axiomatic, enriched and motivic homotopy theory, NATO Sci. Ser. II Math. Phys. Chem., vol. 131, Kluwer Acad. Publ., Dordrecht, 2004, pp. 133–171. MR**2061854**, DOI 10.1007/978-94-007-0948-5_{5} - Paolo Salvatore and Nathalie Wahl,
*Framed discs operads and Batalin-Vilkovisky algebras*, Q. J. Math.**54**(2003), no. 2, 213–231. MR**1989873**, DOI 10.1093/qjmath/54.2.213 - Alexander A. Voronov,
*Notes on universal algebra*, Graphs and patterns in mathematics and theoretical physics, Proc. Sympos. Pure Math., vol. 73, Amer. Math. Soc., Providence, RI, 2005, pp. 81–103. MR**2131012**, DOI 10.1090/pspum/073/2131012

## Additional Information

**Tarje Bargheer**- Affiliation: Department of Mathematics and Statistics, University of Melbourne, Parkville VIC 3010, Australia
- Address at time of publication: Mathematical Sciences Institute, Australian National University, Canberra ACT 0200, Australia
- Email: bargheer@math.ku.dk
- Received by editor(s): March 1, 2012
- Received by editor(s) in revised form: August 22, 2012
- Published electronically: March 31, 2014
- Additional Notes: The author was supported by a postdoctoral grant from the Carlsberg Foundation
- © Copyright 2014
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**366**(2014), 4209-4241 - MSC (2010): Primary 55P50, 18D50
- DOI: https://doi.org/10.1090/S0002-9947-2014-05946-1
- MathSciNet review: 3206457