On MU-homology of connective models of higher Real K-theories

We use the slice filtration to study the $MU$-homology of the fixed points of connective models of Lubin–Tate theory studied by Hill–Hopkins–Ravenel and Beaudry–Hill–Shi–Zeng. We show that, unlike their periodic counterparts $E{O}_n$, the $MU$ homology of $B{P}^{((G))}⟨m{⟩}^G$ usually fails to be even and torsion free. This can only happen when the height $n=m\mathrm{\mid }G\mathrm{\mid }\mathrm{/}2$ is less than $3$, and in the edge case $n=2$, we show that this holds for $tm{f}_0(3)$ but not for $tm{f}_0(5)$, and we give a complete computation of the $M{U}_{\ast }MU$-comodule algebra $M{U}_{\ast }tm{f}_0(3)$.