In this thesis, we calculate the cooperations algebra for the second truncated Brown-Peterson spectrum at the prime 2. We accomplish this through the Adams spectral sequence. We begin by introducing a filtration and a splitting of the mod 2 homology for the second Brown-Peterson spectrum. From this splitting we derive a splitting on the second page of the Adams spectral sequence. This is a splitting into torsion concentrated in Adams filtration 0 and a torsion free component. We then show that this algebraic splitting lifts the stable homotopy category.
After establishing our general structural results, we turn to developing an inductive procedure for determining a basis for the Ext groups after inverting the element detecting 2. From this we determine a basis for the second page of the Adams spectral sequence modulo torsion.