Title: Spectra of BP-linear relations, $v_n$-series, and
BP cohomology of Eilenberg--Mac Lane spaces
Author: Hirotaka Tamanoi
Address: Department of Mathematics
University of California
Santa Cruz, CA 95064
Email: tamanoi@math.ucsc.edu
Abstract: 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 $p$ 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 $p$ 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 $v_n$-series which
are $v_n$-analogues of the $p$-series. Finally we show that the above
main relations come from the $v_n$-series.