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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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