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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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References:
[1] 10.2140/gt.2010.14.1569 · Zbl 1201.53087
[2] 10.4310/JSG.2012.v10.n1.a3 · Zbl 1298.53092
[3] 10.1017/S0305004103007138 · Zbl 1056.55006
[4] 10.1090/conm/504/09875
[5] 10.2140/gt.2013.17.733 · Zbl 1267.19001
[6] 10.1112/jtopol/jtr003 · Zbl 1236.19001
[7] 10.2140/gt.2012.16.1053 · Zbl 1282.19004
[8] 10.1007/s00208-002-0362-0 · Zbl 1025.55005
[9] 10.1016/j.aim.2005.11.004 · Zbl 1155.55302
[10] 10.1016/j.jpaa.2004.11.004 · Zbl 1076.55003
[11] 10.1155/S1073792895000110 · Zbl 0827.18004
[12] 10.1016/j.jpaa.2003.10.030 · Zbl 1060.16010
[13] ; Kontsevich, Conférence Moshé Flato, I. Conférence Moshé Flato, I. Math. Phys. Stud., 21, 255, (2000)
[14] 10.1016/j.jpaa.2006.10.002 · Zbl 1123.18010
[15] 10.1515/9781400830558 · Zbl 1175.18001
[16] 10.1007/978-1-4612-9839-7
[17] 10.1112/S0024611501012692 · Zbl 1017.55004
[18] 10.1090/surv/132
[19] 10.1090/conm/293/04948
[20] 10.1353/ajm.2004.0038 · Zbl 1064.55008
[21] 10.1017/CBO9781107261457 · Zbl 1317.18001
[22] 10.2140/agt.2003.3.287 · Zbl 1028.55013
[23] 10.1016/S0040-9383(02)00006-X · Zbl 1013.55005
[24] ; Seidel, Proceedings of the International Congress of Mathematicians, II, 351, (2002)
[25] 10.1007/s40062-018-0199-1 · Zbl 1432.55018
[26] 10.1023/B:MATH.0000017651.12703.a1 · Zbl 1048.18007
[27] 10.1007/978-94-015-1276-3
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