Topological Hochschild homology and cohomology of \(A_{\infty }\) ring spectra
The topological Hochschild (co)homology \(\text{THH}(A)\) of an \(A_\infty\) ring spectrum \(A\) can be defined after replacing \(A\) with a weakly equivalent \(S\)-algebra \(\widetilde A\). But, it is hard to see how the \(A_\infty\) structure on \(A\) affects the \(S\)-algebra structure on \(\widetilde A\). The author first gives an alternative definition of \(\text{THH}(A)\) in terms of associahedra and cyclohedra. Then the author gives an obstruction theory for endowing a spectrum with an \(A_\infty\) structure, and show that a large class of spectra can be given an \(A_\infty\) structure. Next the author defines two notions of commutativity for an \(A_n\) structure on a spectrum. It turns out that these notions are important to study \(\text{THH}(A)\). Finally, the author gives some sample calculations. The author also compares THH with Morava \(K\)-theory over different ground rings.
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