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 $b{ \rm o}^* tmf$.
Categories: math.AT