[Paper Review] Witt Vectors and Equivariant Ring Spectra

Morten Brun|ArXiv.org|Nov 25, 2004

Homotopy and Cohomology in Algebraic Topology17 references5 citations

TL;DR

This paper establishes a canonical ring homomorphism from the ring of $G$-typical Witt vectors of the zeroth homotopy group of an $E_{ u}$-ring spectrum $T$ with $G$-action to the zeroth homotopy group of the $G$-fixed point spectrum $T^G$. The key result is that for the periodic unitary cobordism spectrum $MP$, this homomorphism is injective, providing a Witt vector structure on equivariant unitary cobordism rings.

ABSTRACT

This paper establishes a connection between equivariant ring spectra and Witt vectors in the sense of Dress and Siebeneicher. Given a commutative ringspectrum T in the highly structured sense, that is, an E-infinity-ringspectrum, with action of a finite group G we construct a ringhomomorphism from the ring of G-typical Witt vectors of the zeroth homotopy group of T to the zeroth homotopy group of the G-fixed point spectrum of T. In the particular case, where T is the periodic unitary cobordism spectrum introduced by Strickland, we show that this ringhomomorphism is injective, and we interpret this in terms of equivariant cobordism.

Motivation & Objective

• To establish a natural ring homomorphism from $\mathbb{W}_G(\pi_0(T))$ to $\pi_0(T^G)$ for $E_{\infty}$-ring spectra $T$ with finite group $G$-action.
• To interpret this homomorphism in terms of equivariant cobordism, particularly for the periodic unitary cobordism spectrum $MP$.
• To show that the homomorphism $\tau_{\widetilde{MP}}$ is injective, though not surjective, for $G$-equivariant $MP$.
• To generalize the classical Witt vector construction to the equivariant stable homotopy category using Tambara functors.
• To provide a categorical framework using bimonoidal categories and partial pseudo-functors to model the interaction between smash products and wedge sums in equivariant spectra.

Proposed method

• Construct a Tambara functor $\widetilde{T}$ from an $E_{\infty}$-ring spectrum $T$ with $G$-action via $[\Sigma^\infty X_+, T]_G$ for finite $G$-sets $X$.
• Apply Dress and Siebeneicher's theory of $G$-typical Witt vectors to the Tambara functor $\widetilde{T}$, yielding a ring homomorphism $\tau_{\widetilde{T}}: \mathbb{W}_G(\pi_0(T)) \to \pi_0(T^G)$.
• Use Laplaza's coherence theory for bimonoidal categories, reformulated via partial pseudo-functors, to ensure compatibility between smash products and wedge sums.
• Analyze the smash-induction of spectra using orthogonal spectra, focusing on the $G$-th smash power $X^{\wedge G}$ with permutation action.
• Leverage homotopical control of cofibrant resolutions and push-out preservation under monoidal functors to ensure stability of constructions.
• Verify that the construction applies to orthogonal spectra, equivariant symmetric spectra, and equivariant $\Gamma$-spaces via stable equivalences.

Experimental results

Research questions

• RQ1 How can the ring of $G$-typical Witt vectors be naturally associated to an equivariant $E_{\infty}$-ring spectrum?
• RQ2 What is the relationship between the $G$-fixed point spectrum of an $E_{\infty}$-ring spectrum and its Witt vector construction?
• RQ3 Is the induced homomorphism $\tau_{\widetilde{T}}: \mathbb{W}_G(\pi_0(T)) \to \pi_0(T^G)$ injective or surjective for specific spectra like $MP$?
• RQ4 Can the classical isomorphism $\tau_{\widetilde{\mathbb{S}}}$ for the sphere spectrum be extended to other equivariant ring spectra?
• RQ5 How does the equivariant unitary cobordism ring $\mathcal{U}^G_*$ relate to the Witt vector construction on the non-equivariant cobordism ring $\mathcal{U}_*$?

Key findings

• The homomorphism $\tau_{\widetilde{T}}: \mathbb{W}_G(\pi_0(T)) \to \pi_0(T^G)$ is constructed canonically for every $E_{\infty}$-ring spectrum $T$ with $G$-action.
• For the sphere spectrum $\mathbb{S}$, the map $\tau_{\widetilde{\mathbb{S}}}$ is an isomorphism, recovering the classical result of Dress and Siebeneicher.
• For topological Hochschild homology $\mathrm{THH}(R)$ with $G$ cyclic, $\tau_{\widetilde{\mathrm{THH}(R)}}$ is also an isomorphism.
• For the periodic unitary cobordism spectrum $MP$, the map $\tau_{\widetilde{MP}}$ is injective but not surjective, even for $G$ cyclic of prime order.
• The ring $\mathbb{W}_G(\mathcal{U}_*)$ embeds as a subring of the equivariant unitary cobordism ring $\mathcal{U}^G_*$, providing a Witt vector structure on equivariant cobordism.
• The construction is stable under change of model structure: it applies to orthogonal spectra, equivariant symmetric spectra, and equivariant $\Gamma$-spaces via stable equivalences.