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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Spectra of BP-linear relations, $v_n$-series,
and BP cohomology of Eilenberg-Mac Lane spaces


Author: Hirotaka Tamanoi
Journal: Trans. Amer. Math. Soc. 352 (2000), 5139-5178
MSC (1991): Primary 55N10, 55N20
Published electronically: July 26, 1999
MathSciNet review: 1661270
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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.


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Additional Information

Hirotaka Tamanoi
Affiliation: Department of Mathematics, University of California at Santa Cruz, Santa Cruz, California 95064
Email: tamanoi@math.ucsc.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02484-8
PII: S 0002-9947(99)02484-8
Keywords: Brown-Peterson (co)homology theory, BP fundamental class, BP topology, Eilenberg--Mac Lane spaces, Milnor primitives, $\Omega $-spectrum, Steenrod algebra, Sullivan exact sequence, $v_{n}$-series
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
Article copyright: © Copyright 1999 American Mathematical Society