The homological slice spectral sequence in motivic and Real bordism

For a motivic spectrum E∈SH(k), let Γ(E) denote the global sections spectrum, where E is viewed as a sheaf of spectra on Sm_k. Voevodsky's slice filtration determines a spectral sequence converging to the homotopy groups of Γ(E). In this paper, we introduce a spectral sequence converging instead to the mod 2 homology of Γ(E) and study the case E=BPGL〈m〉 for k=R in detail. We show that this spectral sequence contains the A_⁎-comodule algebra A_⁎□_A(m)⁎F₂ as permanent cycles, and we determine a family of differentials interpolating between A_⁎□_A(0)⁎F₂ and A_⁎□_A(m)⁎F₂. Using this, we compute the spectral sequence completely for m≤3. In the height 2 case, the Betti realization of BPGL〈2〉 is the C₂-spectrum BP_R〈2〉, a form of which was shown by Hill and Meier to be an equivariant model for tmf₁(3). Our spectral sequence therefore gives a computation of the comodule algebra H_⁎tmf₀(3). As a consequence, we deduce a new (2-local) Wood-type splitting tmf∧X≃tmf₀(3) of tmf-modules predicted by Davis and Mahowald, for X a certain 10-cell complex.