On the spectrum $b{\rm o} \wedge tmf$

TL;DR

This paper constructs a splitting of the spectrum bo ∧ tmf to analyze its algebraic structure, extending previous work on bo-resolutions and providing new insights into the bo_* and bo^* structures of tmf.

Contribution

It introduces a novel splitting of bo ∧ tmf, enabling detailed analysis of bo_*tmf and bo^*tmf, advancing understanding of their algebraic properties.

Findings

Successful splitting of bo ∧ tmf spectrum.

Detailed description of bo_*tmf algebra structure.

Explicit computation of bo^*tmf.

Abstract

M. Mahowald, in his work on $b{\rm o}$-resolutions, constructed a $b{\rm o}$-module splitting of the spectrum $b{\rm o} \wedge b{\rm o}$ into a wedge of summands related to integral Brown-Gitler spectra. In this paper, a similar splitting of $b{\rm o} \sm tmf$ is constructed. This splitting is then used to understand the $b{\rm o}_*$-algebra structure of $b{\rm o}_* tmf$ and allows for a description of $b{\rm o}^* tmf$.