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

Christian Carrick, Michael A. Hill

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 $MU_*tmf_0(3)$.

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

Abstract

We use the slice filtration to study the -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 , the homology of usually fails to be even and torsion free. This can only happen when the height is less than , and in the edge case , we show that this holds for but not for , and we give a complete computation of the -comodule algebra .

Paper Structure

This paper contains 12 sections, 24 theorems, 52 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The $BP$-based slice spectral sequence
  3. Comparison with the $i_*H{\mathbb F}_2$-based slice spectral sequence for $G=C_2$
  4. Odd classes in $BP_*BP^{((G))}\langle m\rangle^G$ for $|G|>2$
  5. The $BP$-homology of $BP_{\mathbb R}\langle n\rangle^{C_2}$
  6. The $E_3$ page and $BP_*ko$
  7. The $E_7$-page
  8. The $n>2$ case and odd classes
  9. The $BP$ homology of $tmf_0(3)$
  10. The relative Adams spectral sequence for $R_0(3)$
  11. The relative Hurewicz map and transfer classes
  12. The underlying $C_2$-action and the image of the restriction map

Key Result

Theorem 1.1

For $|G|>2$ and $m>0$, the $MU$-homology of $BP^{((G))}\langle m\rangle^G$ has torsion classes in odd degrees. When $G=C_2$, the $MU$-homology of $BP_{\mathbb R}\langle m\rangle^{C_2}$ is torsion free and concentrated in even degrees for $m\le 2$ and has torsion classes in odd degrees for $m>2$.

Figures (1)

  • Figure 1: The $i_*BP$-based HSSS for $BP^{((C_4))}\langle 1\rangle$ with transfers in red.

Theorems & Definitions (44)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Corollary 2.5
  • ...and 34 more