## Spectra of $\text {BP}$-linear relations, $v_n$-series, and $\text {BP}$ cohomology of Eilenberg-Mac Lane spaces

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- by Hirotaka Tamanoi
- Trans. Amer. Math. Soc.
**352**(2000), 5139-5178 - DOI: https://doi.org/10.1090/S0002-9947-99-02484-8
- Published electronically: July 26, 1999
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## 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.## References

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## Bibliographic Information

**Hirotaka Tamanoi**- Affiliation: Department of Mathematics, University of California at Santa Cruz, Santa Cruz, California 95064
- Email: tamanoi@math.ucsc.edu
- Received by editor(s): April 30, 1998
- Published electronically: July 26, 1999
- Additional Notes: This research was partially supported by a Faculty Research Grant, University of California at Santa Cruz
- © Copyright 1999 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**352**(2000), 5139-5178 - MSC (1991): Primary 55N10, 55N20
- DOI: https://doi.org/10.1090/S0002-9947-99-02484-8
- MathSciNet review: 1661270