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Higher string topology on general spaces. (English) Zbl 1103.55008

Let \(X\) be a finite-dimensional simplicial complex with a simplicial embedding in a sphere \(S^N\) for some triangulation of the sphere. The author constructs a parametrized spectrum \(SX\) over \(X\), generalizing the construction of Spivak spherical bundles for Poincaré duality spaces, and a spectrum \(\text{Maps}(S^k, X)^{SX}\) so that for \(k\geq 1\), \(C_*(\text{Maps}(S^k, X )^{SX})\) is naturally homotopy equivalent to a \({\mathcal C}_{k+1}\)-algebra in the category of chain complexes. Here \(C_*\) denotes chain complexes with coefficients in a field and \({\mathcal C}_k\) denotes the unframed little disk operad. The method of the proof of this result shows that \(C_*(\text{Maps}(S^k, X)^{SX})\) is naturally equivalent to the Hochschild complex in the category of \({\mathcal C}_k\)-algebras. This result gives a generalized analogue of the Gerstenhaber algebra defined by M. Chas and D. Sullivan in their seminal paper on string topology.
In the last section the author relates her result to the Ginzburg-Kapranov Koszul duality.

MSC:

55P48 Loop space machines and operads in algebraic topology
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
55N45 Products and intersections in homology and cohomology
18D50 Operads (MSC2010)
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