The Segal conjecture for topological Hochschild homology of complex cobordism

TL;DR

This paper proves a p-adic equivalence between fixed and homotopy fixed point spectra of topological Hochschild homology for complex cobordism spectra, extending the classical Segal conjecture to a new topological context.

Contribution

It introduces a topological Singer construction and relates it to the Tate construction, generalizing the Segal conjecture to topological Hochschild homology of complex cobordism spectra.

Findings

Fixed and homotopy fixed point spectra are p-adically equivalent for MU and BP.

The approach relates the Tate construction to a topological Singer construction.

Generalizes the classical Segal conjecture to a topological setting.

Abstract

We study the C_p-equivariant Tate construction on the topological Hochschild homology THH(B) of a symmetric ring spectrum B by relating it to a topological version R_+(B) of the Singer construction, extended by a natural circle action. This enables us to prove that the fixed and homotopy fixed point spectra of THH(B) are p-adically equivalent for B = MU and BP. This generalizes the classical C_p-equivariant Segal conjecture, which corresponds to the case B = S.