The homological slice spectral sequence in motivic and Real bordism

For a motivic spectrum $E\in \mathrm{SH}\left(k\right)$, let $\Gamma \left(E\right)$ denote the global sections spectrum, where E is viewed as a sheaf of spectra on ${\mathrm{Sm}}_k$. Voevodsky's slice filtration determines a spectral sequence converging to the homotopy groups of $\Gamma \left(E\right)$. In this paper, we introduce a spectral sequence converging instead to the mod 2 homology of $\Gamma \left(E\right)$ and study the case $E=BPGL〈m〉$ for $k=R$ in detail. We show that this spectral sequence contains the $A_{⁎}$-comodule algebra $A_{⁎}{\square }_{A{\left(m\right)}_{⁎}}{F}_2$ as permanent cycles, and we determine a family of differentials interpolating between $A_{⁎}{\square }_{A{\left(0\right)}_{⁎}}{F}_2$ and $A_{⁎}{\square }_{A{\left(m\right)}_{⁎}}{F}_2$. Using this, we compute the spectral sequence completely for $m\le 3$.

In the height 2 case, the Betti realization of $BPGL〈2〉$ is the $C_2$-spectrum $B{P}_R〈2〉$, a form of which was shown by Hill and Meier to be an equivariant model for ${\mathrm{tmf}}_1\left(3\right)$. Our spectral sequence therefore gives a computation of the comodule algebra $H_{⁎}{\mathrm{tmf}}_0\left(3\right)$. As a consequence, we deduce a new (2-local) Wood-type splitting$\mathrm{tmf}\wedge X\simeq {\mathrm{tmf}}_0\left(3\right)$ of tmf-modules predicted by Davis and Mahowald, for X a certain 10-cell complex.