The Bp -theory of Two-fold Products of Projective Spaces

In [KWa, KWb] the need for the Brown-Peterson cohomology (for p = 2) of a product of two real projective spaces arose. In particular, they needed to understand the elements not in the tensor product and how they behaved under maps. Although quick computations with the Adams spectral sequence or the AtiyahHirzebruch spectral sequence suggest the answer, there seemed to be nothing explicit enough in the literature, but much of what we do is well known. We always use reduced cohomology. Recall that BP ∗ ' Z(2)[v1, v2, . . .]. Let x ∈ BP (RP ) be the standard generator coming from BP ∗(CP∞). The required theorems are as follows: