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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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The cleavage operad and string topology of higher dimension
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by Tarje Bargheer PDF
Trans. Amer. Math. Soc. 366 (2014), 4209-4241 Request permission


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:
  • 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:
  • MathSciNet review: 3206457