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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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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

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