Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-99-02484-8
Published electronically: July 26, 1999
MathSciNet review: 1661270
Full-text PDF

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.


References [Enhancements On Off] (What's this?)

  • [Ad] J. F. Adams, Stable homotopy and generalized homology, University of Chicago Press, Chicago, Illinois, 1974. MR 53:6534
  • [Ar] S. Araki, Typical formal groups in complex cobordism and K-theory, Lecture Notes Math., Kyoto Univ., vol. 6, Kinokuniya Book Store, 1973. MR 51:11549
  • [Ba] N. A. Baas, On bordism theory of manifolds with singularity, Math. Scand. 33 (1973), 279-302. MR 49:11547b
  • [BM] N. A. Baas and Ib Madsen, On the realization of certain modules over the Steenrod algebra, Math. Scand. 31 (1971), 220-224. MR 51:14048
  • [H] M. Hazewinkel, Formal groups and applications, Academic Press, New York, 1978. MR 82a:14020
  • [JW1] D. C. Johnson and W. S. Wilson, Projective dimension and Brown-Peterson homology, Topology 12 (1973), 327-353. MR 48:12576
  • [JW2] D. C. Johnson and W. S. Wilson, BP operations and Morava's extraordinary K-theories, Math. Z. 144 (1975), 55-75. MR 51:14025
  • [JW3] D. C. Johnson and W. S. Wilson, The projective dimension of the complex cobordism of Eilenberg-MacLane spaces, Osaka J. Math. 14 (1977), 533-536. MR 57:7584
  • [JW4] D. C. Johnson and W. S. Wilson, The Brown-Peterson homology of elementary $p$-groups, Amer. J. Math. 107 (1985), 427-453. MR 86j:55008
  • [JY] D. C. Johnson and Z. Yoshimura, Torsion in Brown-Peterson homology and Hurewicz homomorphisms, Osaka J. Math. 17 (1980), 117-136. MR 81b:55010
  • [L] P. S. Landweber, Coherence, flatness and cobordism of classifying spaces, Proceedings of Advanced Study Institute on Algebraic Topology, Aarhus, 1970, pp. 256-269. MR 42:6845
  • [M1] J. W. Milnor, The Steenrod algebra and its dual, Ann. of Math. 67 (1958), 150-171. MR 20:6092
  • [M2] J. W. Milnor, On the cobordism ring $\Omega ^{*}$ and a complex analogue, Amer. J. Math. 82 (1960), 505-521. MR 22:9975
  • [M3] J. W. Milnor, On axiomatic homology theory, Pacific J. Math. 12 (1962), 337-341. MR 28:2544
  • [Mo] J. Morava, A product for the odd-primary bordism of manifolds with singularities, Topology 18 (1979), 177-186. MR 80k:57063
  • [Q] D. Quillen, On the formal group laws of unoriented and complex cobordism theory, Bull. Amer. Math. Soc. 75 (1969), 1293-1298. MR 40:6565
  • [R1] D. C. Ravenel, Complex cobordism and stable homotopy groups of spheres, Academic Press, Orlando, FL, 1986. MR 87j:55003
  • [R2] D. C. Ravenel, Nilpotence and Periodicity in Stable Homotopy Theory, Annals of Math. Studies 128, Princeton Univ. Press, Princeton, NJ, 1992. MR 94b:55015
  • [RW1] D. C. Ravenel and W. S. Wilson, The Hopf ring for complex cobordism, J. Pure Appl. Algebra 9 (1977), 241-280. MR 56:6644
  • [RW2] D. C. Ravenel and W. S. Wilson, The Morava K theories of Eilenberg-MacLane spaces and the Conner-Floyd conjecture, Amer. J. Math. 102 (1980), 691-748. MR 81i:55005
  • [RWY] D. C. Ravenel, W. S. Wilson, and N. Yagita, Brown-Peterson cohomology from Morava $K$-theory, $K$-theory 15 (1998), 147-199. CMP 99:02
  • [S] K. Sinkinson, The cohomology of certain spectra associated with th Brown-Peterson spectrum, Duke Math. J. 43 (1976), 605-622. MR 53:14471
  • [T1] H. Tamanoi, The image of the BP Thom map for Eilenberg-Mac Lane spaces, Trans. AMS 349 (1997), 1209-1237. MR 97i:55012
  • [T2] H. Tamanoi, $Q$-subalgebras, Milnor basis, and cohomology of Eilenberg-Mac Lane spaces, To appear in J. Pure and Applied Algebra.
  • [W] W. S. Wilson, The $\Omega $-spectrum for Brown-Peterson cohomology, Part I, Comment. Math. Helv. 48 (1973), 45-55; Part II, Amer. J. Math. 97 (1975), 101-123.MR 48:5055; MR 52:4271
  • [Y1] N. Yagita, On relations between Brown-Peterson cohomology and the ordinary mod $p$ cohomology theory, Kodai Math. J. 7 (1984), 273-285. MR 85g:55007
  • [Y2] N. Yagita, On the image $\rho \bigl (BP^{*}(X) \rightarrow H^{*}(X;\mathbb{Z}_{p})\bigr )$, Advanced Studies in Pure Math. 9 (1986, Homotopy Theory and Related Topics), 335-344. MR 88j:55005
  • [Z] R. Zahler, The Adams-Novikov spectral sequence for the spheres, Ann. of Math. 96 (1972), 480-504. MR 47:7742

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 55N10, 55N20

Retrieve articles in all journals with MSC (1991): 55N10, 55N20


Additional Information

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

DOI: https://doi.org/10.1090/S0002-9947-99-02484-8
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

American Mathematical Society