On the BP homology and cohomology of \(P^{2n}\wedge P^{2m}\) (Q752539)

Let BP be the Brown-Peterson spectrum associated with the prime 2. Then Landweber's Künneth short exact sequence gives \(BP^*(P^{2n}\wedge P^{2m})\) as a direct sum of Tor and tensor product modules of \(BP^*(P^{2n})\) and \(BP^*(P^{2m})\) over \(BP^*\). The tensor product module is well understood. The author computes the Tor groups as \(BP^*\) modules and describes the \(v_ 1\)-torsion submodule of the tensor product term. The dual case of the homology is also considered.