On the Hopf ring for

Kriz and Hu construct a real Johnson–Wilson spectrum, ER ( n ) , which is 2 n + 2 ( 2 n − 1 ) periodic. ER ( 1 ) is just KO ( 2 ) . We do two things in this paper. First, we compute the homology of the 2 n − 1 spaces ER ( n ) ̲ 2 n + 2 k in the Omega spectrum for ER ( n ) . It turns out the double of these Hopf algebras gives the homology Hopf algebras for the even spaces for E ( n ) . As a byproduct of this we get the homology of the zeroth spaces for the Omega spectrum for real complex cobordism and real Brown–Peterson cohomology. The second result is to compute the homology Hopf ring for all 48 spaces in the Omega spectrum for ER ( 2 ) . This turns out to be generated by very few elements.