String topology and the based loop space

Abstract

For M a closed, connected, oriented manifold, we obtain the Batalin-Vilkovisky (BV) algebra of its string topology through homotopy-theoretic constructions on its based loop space. In particular, we show that the Hochschild cohomology of the chain algebra C_*ΩM carries a BV algebra structure isomorphic to that of the loop homology ${\mathbb{H}}_{\ast }(LM)$. Furthermore, this BV algebra structure is compatible with the usual cup product and Gerstenhaber bracket on Hochschild cohomology. To produce this isomorphism, we use a derived form of Poincaré duality with C_*ΩM-modules as local coefficient systems, and a related version of Atiyah duality for parametrized spectra connects the algebraic constructions to the Chas-Sullivan loop product.

38 pages. Condensed version of the author's Stanford University Ph.D. thesis

References in corpus (10)

• Algebras and modules in monoidal model categories
• Calabi-Yau algebras
• Batalin-Vilkovisky algebras and two-dimensional topological field theories
• Equivalences of monoidal model categories
• Homotopy limits and colimits and enriched homotopy theory
• The string topology BV algebra, Hochschild cohomology and the Goldman bracket on surfaces
• String topology of Poincare duality groups
• A Homotopy Theoretic Proof of the BV Identity in Loop Homology
• Van Den Bergh isomorphisms in String Topology
• Equivariant operads, string topology, and Tate cohomology

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• Continuous and coherent actions on wrapped Fukaya categories
• An ${\mathsf{A}}_{\infty }$ version of the Poincaré lemma
• Koszul A-infinity algebras and free loop space homology
• Morse homology with DG coefficients
• On Cohen-Jones isomorphism in string topology
• Flat Z-graded connections and loop spaces