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Mathematics > Algebraic Topology

arXiv:1103.6198 (math)
[Submitted on 31 Mar 2011]

Title:String topology and the based loop space

Authors:Eric J. Malm
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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_*\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 Poincaré 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.
Comments: 38 pages. Condensed version of the author's Stanford University Ph.D. thesis
Subjects: Algebraic Topology (math.AT)
MSC classes: 55P50, 16E30, 55U30
Cite as: arXiv:1103.6198 [math.AT]
  (or arXiv:1103.6198v1 [math.AT] for this version)
  https://doi.org/10.48550/arXiv.1103.6198

Submission history

From: Eric Malm [view email]
[v1] Thu, 31 Mar 2011 14:43:58 UTC (41 KB)
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