Algebra/Topology seminar

In this talk, I will explain how to compute the C_3-equivariarnt relative Adams spectral sequence for the Borelification of tmf(2).This yields an entirely algebraic computation of the 3-local homotopy groups of tmf. The final answer is well-known of course -- the novelty here is that the rASS is completely determined by its E_1-page as a chain complex of Mackey functors. Explicitly, the input consists of the Hopf algebroid structure on F_3 \otimes_{tmf(2)}  F_3   modulo transfer, which is deduced from the structure maps on the equivariant dual Steenrod algebra, as well as the knowledge of the homotopy groups of tmf(2) along with the C_3-action.  Then we construct a bifiltration on tmf(2) and use synthetic arguments to deduce the Adams differentials from the associated square of spectral sequences. The rASS degenerates on E_{12} for tridegree reasons and stabilizes to a periodic pattern that essentially lies within a band of slope 1/4. This is joint work with Jeremy Hahn, Andrew Senger, and Foling Zou.