A multiplicative comparison of MacLane homology and topological Hochschild homology

TL;DR

This paper proves a conjecture relating MacLane homology and topological Hochschild homology by constructing an equivalence of certain spectra, providing new proofs and extending the known relationships in algebraic topology.

Contribution

It constructs a symmetric monoidal equivalence between MacLane's $Q$-construction and the tensor product with integers, confirming a conjecture and offering new proofs of homology equivalences.

Findings

Established an equivalence $Q(R) \,\simeq\, \mathbb Z \otimes R$ in $A_infty$ ring spectra

Provided a symmetric monoidal proof that $ ext{HML}(R,M) \simeq \text{THH}(R,M)$

Extended the equivalence to $E_\infty$ ring spectra for commutative rings

Abstract

Let $Q$ denote MacLane's $Q$-construction, and $\otimes$ denote the smash product of spectra. In this paper we construct an equivalence $Q(R)\simeq \mathbb Z\otimes R$ in the category of $A_\infty$ ring spectra for any ring $R$, thus proving a conjecture made by Fiedorowicz, Schw\"anzl, Vogt and Waldhausen in "MacLane homology and topological Hochschild homology". More precisely, we construct is a symmetric monoidal structure on $Q$ (in the $\infty$-categorical sense) extending the usual monoidal structure, for which we prove an equivalence $Q(-)\simeq \mathbb Z\otimes -$ as symmetric monoidal functors, from which the conjecture follows immediately. From this result, we obtain a new proof of the equivalence $\mathrm{HML}(R,M)\simeq \mathrm{THH}(R,M)$ originally proved by Pirashvili and Waldaushen in "MacLane homology and topological Hochschild homology" (a different paper from the one cited above). This equivalence is in fact made symmetric monoidal, and so it also provides a proof of the equivalence $\mathrm{HML}(R)\simeq \mathrm{THH}(R)$ as $E_\infty$ ring spectra, when $R$ is a commutative ring.