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$$E_2$$ structures and derived Koszul duality in string topology. (English) Zbl 1426.55011
R. L. Cohen and J. D. S. Jones [Math. Ann. 324, No. 4, 773–798 (2002; Zbl 1025.55005)] have given a homotopical interpretation of the loop product in string topology due to Chas and Sullivan by relating a certain Thom spectrum of the free loop space $$LM$$ of a closed orientable manifold $$M$$ with the topological Hochschild cohomology $$\text{THC}(DM)$$ of the Spanier-Whitehead dual $$DM$$. The homology of $$\text{THC}(DM)$$ is canonically isomorphic to the Hochschild cohomology $$HH^*(C^*(M))$$ of the cochain algebra $$C^*(M)$$. Cohen-Jones in [loc. cit.] produced an isomorphism between the loop homology of $$M$$ with the loop product and the Hochschild cohomology with the cup product.
In [J. Pure Appl. Algebra 199, No. 1–3, 43–59 (2005; Zbl 1076.55003)], Y. Félix et al. have constructed an isomorphism of Gerstenhaber algebras $$HH^*(C^*(M))\cong HH^*(\overline{\Omega}C_*(M))$$ when $$M$$ is simply connected, where $$\overline\Omega C_*(M)$$ denotes the cobar construction. The homology of $$\text{THC}(\Sigma_+^\infty \Omega M)$$ is isomorphic to $$HH^*(\overline{\Omega}C_*(M))$$. Therefore, a spectral model version of the result due to Félix, Menichi and Thomas is expected. In the paper under review, Theorem A shows that for a simply connected finite cell complex $$X$$, the topological Hochschild cohomologies $$\text{THC}(DX)$$ and $$\text{THC}(\Sigma_+^\infty \Omega X)$$ are weakly equivalent as $$E_2$$ ring spectra.
The paper considers a generalization for $$\text{THC}$$ in the setting of small spectral categories which generalize associative ring spectra. The small category is enriched over spectra. More precisely, it has a bi-indexed spectrum as a set of morphisms. The topological Hochschild cohomology of a small spectral category $${\mathcal C}$$ is defined by $$\text{THC}({\mathcal C}) = \text{CC}({\mathcal C}^{\text{Cell}, \Omega})$$ with the Hochschild-Mitchell construction $$\text{CC}$$ and a cofibrant-fibrant replacement $${\mathcal C}^{\text{Cell}, \Omega}$$ of $${\mathcal C}$$ in a topologically enriched closed model category.
Let $${\mathcal C}$$ and $${\mathcal D}$$ be small spectral categories and $${\mathcal M}$$ a $$({\mathcal C}, {\mathcal D})$$-bimodule. Then there are canonical maps in the category of homotopical $$({\mathcal C}, {\mathcal C})$$-bimodules and homotopical $$({\mathcal D}, {\mathcal D})$$-bimodules ${\mathcal C} \to {\mathbb R}\text{Hom}_{{\mathcal D}^{\text{op}}}({\mathcal M}, {\mathcal M}) \ \ \text{and} \ \ {\mathcal D} \to {\mathbb R}\text{Hom}_{{\mathcal C}}({\mathcal M}, {\mathcal M}),$ respectively. By definition, we say that $${\mathcal M}$$ satisfies the double centralizer condition when these both maps are weak equivalences and the single centralizer condition for $${\mathcal D}$$ when the second map (out of $${\mathcal D}$$) is a weak equivalence. Tying in the double centralizer condition with weak equivalences between the Hochschild-Mitchell constructions for certain small spectral categories (Theorem 5.7), the authors deduce the following result which is regarded as the spectral version of the main theorem of B. Keller [“Derived invariance of higher structures on the Hochschild complex”, Preprint, https://webusers.imj-prg.fr/~bernhard.keller/publ/dih.pdf].
Theorem E. Let $${\mathcal C}$$ and $${\mathcal D}$$ be small spectral categories and $${\mathcal M}$$ a $$({\mathcal C}, {\mathcal D})$$-bimodule that satisfies the single centralizer condition for $${\mathcal D}$$, then there exists a canonical map $$\text{THC}({\mathcal C}) \to \text{THC}({\mathcal D})$$ in the homotopy category of $$E_2$$ ring spectra. If $${\mathcal M}$$ satisfies the double centralizer condition, then $$\text{THC}({\mathcal C}) \to \text{THC}({\mathcal D})$$ is an isomorphism in the homotopy category of $$E_2$$ ring spectra.
Theorem A follows from Theorem E as an immediate corollary.
##### MSC:
 55P50 String topology 16D90 Module categories in associative algebras 16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
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