String topology and the based loop space

Details String topology and the based loop space String topology and the based loop space

2011-03-31

arxiv.org/abs/1103.6198v1

Mathematics

Algebraic Topology

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

Scientific paper

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_*\Omega M carries a BV algebra structure isomorphic to that of the loop homology $\mathbb{H}_*(LM)$. 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 Poincar\'e duality with C_*\Omega M-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.

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