A homotopy theoretic realization of string topology
UNSPECIFIED. (2002) A homotopy theoretic realization of string topology. MATHEMATISCHE ANNALEN, 324 (4). pp. 773-798. ISSN 0025-5831Full text not available from this repository.
Official URL: http://dx.doi.org/10.1007/s00208-002-0362-0
Let M be a closed, oriented manifold of dimension d. Let LM be the space of smooth loops in M. In  Chas and Sullivan defined a product on the homology H-* (LM) of degree -d. They then investigated other structure that this product induces, including a Batalin-Vilkovisky structure, and a Lie algebra structure on the S-1 equivariant homology H-*(S 1) (LM). These algebraic structures, as well as others, came under the general heading of the "string topology" of M. In this paper we will describe a realization of the Chas-Sullivan loop product in terms of a ring spectrum structure on the Thom spectrum of a certain virtual bundle over the loop space. We also show that an operad action on the homology of the loop space discovered by Voronov has a homotopy theoretic realization on the level of Thom spectra. This is the "cactus operad" defined in  which is equivalent to operad of framed disks in R-2. This operad action realizes the Chas-Sullivan BV structure on H-*(LM). We then describe a cosimplicial model of this ring spectrum, and by applying the singular cochain functor to this cosimplicial spectrum we show that this ring structure can be interpreted as the cup product in the Hochschild cohomology, HH*(C*(M); C*(M)).
|Item Type:||Journal Article|
|Subjects:||Q Science > QA Mathematics|
|Journal or Publication Title:||MATHEMATISCHE ANNALEN|
|Official Date:||December 2002|
|Number of Pages:||26|
|Page Range:||pp. 773-798|
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