A homotopy theoretic realization of string topology

By R. Cohen, John D. S. Jones. 2001

269 citations recorded by the source corpus.

Abstract. Let M be a closed, oriented manifold of dimension d. Let LM be the space of smooth loops in M. In [2] 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 S1 equivariant homology H*S1(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 [6] which is equivalent to operad of framed disks in . 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)).

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0e57d12c897022cb241c1cafea137fc0c649d6ff

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Citations in the corpus

Lagrangian topology and enumerative geometry (2010, unknown): “[CJ]) between H∗(Λ(L), K) and the Hochschild cohomology HH∗(C ·(L);C ·(L)) where C ·(−) is the singular cochain complex.”
Membrane Topology (2006, background): “Associativity is proved as in the case of string topology [8] .”
Arc Operads and Arc Algebras (2002, methodology): “If we restrict ourselves to this subspace we obtain from the above following [1, 16, 2]: 4.”
The Hochschild cohomology of a closed manifold (2004, background): “There is no complete written proof of this conjecture in the litterature; however Tradler, Cohen, Jones have already understood the situation and have given substantial parts of a proof ([4], [6]).”
Rational string topology (2004, background): “When coefficients are in a field, there is also an isomorphism of graded vector spaces, [5], [10] from the singular homology of the space LM to the Hochschild cohomology of the singular cochain complex of M : H∗(LM)→ HH (CM,CM) .”
Algebraic models in geometry (2008, unknown): “38 ([62], [191], [94]) The isomorphism H∗(LN;Q) ∼= HH∗(A)”
A bordism approach to string topology (2003, background): “The BV -structure onH⁄(LM) comes in fact form a geometric action of the cacti operad [ 12 ], [46] (normalized cacti with spines in the terminology of R. Kaufmann [31]).”
Symplectic cohomology and Viterbo's theorem (2013, methodology): “MR1312580 (95m:58055) [26] Ralph L. Cohen and John D. S. Jones, A homotopy theoretic realization of string topology, Math.”