The homological slice spectral sequence in motivic and Real bordism

Abstract

For a motivic spectrum $E\in \mathcal{SH}(k)$, let $\Gamma(E)$ 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 $\Gamma(E)$. In this paper, we introduce a spectral sequence converging instead to the mod 2 homology of $\Gamma(E)$ and study the case $E=BPGL\langle m\rangle$ for $k=\mathbb R$ in detail. We show that this spectral sequence contains the $\mathcal{A}_*$-comodule algebra $(\mathcal{A}//\mathcal{A}(m))^*$ as permanent cycles, and we determine a family of differentials interpolating between $(\mathcal{A}//\mathcal{A}(0))^*$ and $(\mathcal{A}//\mathcal{A}(m))^*$. Using this, we compute the spectral sequence completely for $m\le 3$. In the height 2 case, the Betti realization of $BPGL\langle 2\rangle$ is the $C_2$-spectrum $BP_{\mathbb R}\langle 2\rangle$, a form of which was shown by Hill and Meier to be an equivariant model for $\mathrm{tmf}_1(3)$. Our spectral sequence therefore gives a computation of the comodule algebra $H_*\mathrm{tmf}_0(3)$. As a consequence, we deduce a new ($2$-local) Wood-type splitting \[\mathrm{tmf}\wedge X\simeq \mathrm{tmf}_0(3)\] of $\mathrm{tmf}$-modules predicted by Davis and Mahowald, for $X$ a certain 10-cell complex.