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

Author:
Hirotaka Tamanoi

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

Published electronically:
July 26, 1999

MathSciNet review:
1661270

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

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