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

TL;DR

This paper investigates the MU-homology of fixed points of connective models of higher Real K-theories, revealing that their homology often fails to be even and torsion free except in low-height cases, with explicit computations for tmf_0(3).

Contribution

It provides new insights into the MU-homology of connective models of Lubin--Tate theory, especially highlighting differences from periodic counterparts and offering explicit computations.

Findings

MU homology often not even and torsion free for these models

Failure occurs when height n=m|G|/2 is less than 3

Complete computation of MU_*tmf_0(3)

Abstract

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 $EO_n$, the $MU$ homology of $BP^{((G))}\langle m\rangle^G$ usually fails to be even and torsion free. This can only happen when the height $n=m|G|/2$ is less than $3$, and in the edge case $n=2$, we show that this holds for $tmf_0(3)$ but not for $tmf_0(5)$, and we give a complete computation of the $MU_*MU$-comodule algebra $MU_*tmf_0(3)$.