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The cleavage operad and string topology of higher dimension. (English) Zbl 1312.55009

The cacti operad was introduced by A. A. Voronov [Proc. Symp. Pure Math. 73, 81–103 (2005; Zbl 1083.18005)] in order to model the Chas-Sullivan product on the homology of the free loop space of a compact manifold. The author defines generalizations of this operad for manifolds \(N\) equipped with an embedding into the euclidean space \(\mathbb R^{n+1}\). The cleavage operad, which is the result of his construction, is a colored topological operad \(\mathrm{Cleav}_N\) which acts on the space of correspondences between \(N\) and any manifold \(M\), and whose elements are associated to composition schemes of hyperplanes used to cleave the manifold \(N\) into smaller pieces.
The author studies the cleavage operad associated to a sphere \(N = S^n\) more closely. He proves that \(\mathrm{Cleav}_{S^n}\) is an \(E_{n+1}\)-operad. He then forms a semi-direct product \(\mathrm{Cleav}_{S^n}\rtimes SO(n+1)\) to get an operad acting on the space of correspondences \(M^{S^n}\) and with the same homology as the operad of framed little \(n\)-disks.

MSC:

55P50 String topology
55P48 Loop space machines and operads in algebraic topology

Citations:

Zbl 1083.18005
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References:

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