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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The cleavage operad and string topology of higher dimension

Author: Tarje Bargheer
Journal: Trans. Amer. Math. Soc. 366 (2014), 4209-4241
MSC (2010): Primary 55P50, 18D50
Published electronically: March 31, 2014
MathSciNet review: 3206457
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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.

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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

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
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.