Spectra of 𝐵𝑃-linear relations, 𝑣_{𝑛}-series, and 𝐵𝑃 cohomology of Eilenberg-Mac Lane spaces

On Brown-Peterson cohomology groups of a space, we introduce a natural inherent topology, BP topology, which is always complete Hausdorff for any space. We then construct a spectra map which calculates infinite BP-linear sums convergent with respect to the BP topology, and a spectrum which describes infinite sum BP-linear relations in BP cohomology. The mod p p cohomology of this spectrum is a cyclic module over the Steenrod algebra with relations generated by products of exactly two Milnor primitives. We show a close relationship between BP-linear relations in BP cohomology and the action of the Milnor primitives on mod p p cohomology. We prove main relations in the BP cohomology of Eilenberg–Mac Lane spaces. These are infinite sum BP-linear relations convergent with respect to the BP topology. Using BP fundamental classes, we define v n v_{n} -series which are v n v_{n} -analogues of the p p -series. Finally, we show that the above main relations come from the v n v_{n} -series.