Topological $ΔG$ homology of rings with twisted $G$-action

We construct topological $ΔG$-homology for rings with twisted $G$-action. Here a ring with twisted $G$-action is a common generalization of a ring with anti-involution and a ring with $G$-action. This construction recovers as special cases topological Hochschild homology (THH) of rings, with its $S^1$-action, and Real topological Hochschild homology (THR) of rings with anti-involution, with its $O(2)$-action. A new example of this construction is quaternionic topological Hochschild homology (THQ) of rings with twisted $C_4$-action, which carries a $Pin(2)$-action. We prove that THQ of a loop space with twisted $C_4$-action can be $Pin(2)$-equivariantly identified with a twisted free loop space. Other new examples of interest are topological symmetric homology and topological hyperoctrahedral homology and more generally topological twisted symmetric homology. We prove a homotopical version of results of Fiedorowicz, Ault, and Graves computing these new topological homology theories on loop spaces with twisted $G$-action. A key step of independent interest in this program is the construction of a new family of crossed simplicial groups, which correspond to operads that encode the structure of rings with twisted $G$-action.

💡 Research Summary

The paper introduces a unifying framework for constructing homology theories of ring spectra equipped with a “twisted G‑action”. A twisted G‑action is a generalization of both a plain G‑action and an anti‑involution: even elements of the group act by ring homomorphisms, odd elements act by anti‑homomorphisms. The authors’ central idea is to use crossed simplicial groups (CSGs) – categorical extensions of the simplex category Δ that incorporate families of automorphism groups – as the combinatorial backbone for these constructions.

First, they develop a new family of CSGs denoted Δ ϕ ≀ Σ, associated to any group homomorphism (parity) ϕ : G → {±1}. This construction interpolates between the classical cyclic, dihedral, quaternionic, symmetric, and hyperoctahedral categories. They prove that any CSG Δ G admits a canonical map to the CSG attached to its parity, i.e. Δ ϕ ≀ Σ (Theorem 2.9, Proposition 2.22). This factorisation supplies a universal source of operadic data for all twisted actions.

Next, they identify twisted G‑rings as algebras over a “twisted operad” Asso c_ϕ (Proposition 3.15). Asso c_ϕ is a semidirect product operad that encodes both ordinary multiplication and the anti‑multiplication coming from odd group elements. They then show that the pointed version Δ ϕ ≀ Σ⁺ is precisely the active part of the category of operators associated to Asso c_ϕ (Theorem 3.17, Remark 3.24). Consequently, symmetric monoidal functors from Δ ϕ ≀ Σ⁺ to spectra are exactly the data needed to turn a twisted G‑ring into a spectrum.

With these ingredients, the authors lift the classical cyclic bar construction to the ∞‑categorical setting for any self‑dual CSG Δ G. The resulting object, denoted TH_G(R), is a spectrum equipped with a canonical action of the topological group |G·| (the geometric realization of the simplicial set of automorphism groups). For the familiar cases Δ C, Δ D, and Δ Q they recover topological Hochschild homology (THH) with its S¹‑action, Real topological Hochschild homology (THR) with its O(2)‑action, and a new invariant THQ (quaternionic topological Hochschild homology) with a Pin(2)‑action.

A major computational achievement is Theorem 5.25, which identifies THQ of a loop space with twisted C₄‑action as the suspension spectrum of a Pin(2)‑equivariant twisted free loop space L_τX. Here L_τX consists of paths γ: