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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
DOI: https://doi.org/10.1090/S0002-9947-2014-05946-1
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
Email: bargheer@math.ku.dk

DOI: https://doi.org/10.1090/S0002-9947-2014-05946-1
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.

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