Examples of disk algebras
Sanath Devalapurkar, Jeremy Hahn, Tyler Lawson, Andrew Senger, Dylan Wilson
TL;DR
The paper develops a framework for enriching key spectra with $Disk_n^B$-algebra structures to enable circle-equivariant Hochschild-type constructions and descent in THH computations. By extending $E_n$-algebras to $Disk_n^B$-algebras using orientability and Thom-spectrum techniques, it constructs explicit refinements for $BP$, $BP<\!n\!>$, Ravenel's $X(n)$, and graded sphere-polynomial rings, ensuring compatible multiplicative structures. The main results provide general criteria for when Thom spectra acquire $Disk_n^B$-algebra structures and illustrate concrete instances, such as $X(n)$ as a $Disk_2^{BU(1)}$-algebra and $BP$ as a $Disk_4^{BU(2)}$-algebra under $MU$, guided by obstruction theory. These refinements deepen the understanding of highly structured multiplicative structures in chromatic homotopy theory and pave the way for refined $S^1$-equivariant THH and cyclotomic analyses.
Abstract
We produce refinements of the known multiplicative structures on the Brown--Peterson spectrum $BP$, its truncated variants $BP\langle n \rangle$, Ravenel's spectra $X(n)$, and evenly graded polynomial rings over the sphere spectrum. Consequently, topological Hochschild homology relative to these rings inherits a circle action.