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On the multiplicative structure of topological Hochschild homology. (English) Zbl 1141.55005
The authors prove that if \(R\) is an \(E_n\)-ring spectrum, then the topological Hochschild homology is an \(E_{n-1}\)-ring spectrum. This is shown by proving that (1) \(\mathcal Ass\otimes\mathcal C_{n-1}\) is an \(E_n\)-operad, and (2) there is a chain \(Y_R\gets X_R\to R\) of \(E_n\)-ring maps that are equivalences of spectra and such that \(Y_R\) is an \(\mathcal Ass\otimes\mathcal C_{n-1}\)-algebra where \(\mathcal Ass\) is the associative operad and \(\mathcal C_{n-1}\) is the little \(n-1\)-cube operad. One then gets a chain \(THH(Y_R)\gets THH(X_R)\to THH(R)\) of equivalences where \(THH(Y_R)\) is manifestly an \(E_{n-1}\)-ring spectrum. The proof of (1) uses a variant of Berger’s idea for a cellular decomposition of \(\mathcal C_{n-1}\) to describe \(\mathcal Ass\otimes\mathcal C_{n-1}\). The proof of (2) follows from this by a monoadic bar construction argument. That \(THH\) of an \(E_n\)-ring spectrum is an \(E_{n-1}\)-ring spectrum has been claimed independently by Basterra and Mandell.

MSC:
55P43 Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.)
18D50 Operads (MSC2010)
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