Spectra of BPlinear relations, series, and BP cohomology of EilenbergMac Lane spaces
Author:
Hirotaka Tamanoi
Journal:
Trans. Amer. Math. Soc. 352 (2000), 51395178
MSC (1991):
Primary 55N10, 55N20
Published electronically:
July 26, 1999
MathSciNet review:
1661270
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Abstract: On BrownPeterson 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 BPlinear sums convergent with respect to the BP topology, and a spectrum which describes infinite sum BPlinear relations in BP cohomology. The mod 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 BPlinear relations in BP cohomology and the action of the Milnor primitives on mod cohomology. We prove main relations in the BP cohomology of EilenbergMac Lane spaces. These are infinite sum BPlinear relations convergent with respect to the BP topology. Using BP fundamental classes, we define series which are analogues of the series. Finally, we show that the above main relations come from the series.
 [Ad]
J.
F. Adams, Stable homotopy and generalised homology, University
of Chicago Press, Chicago, Ill.London, 1974. Chicago Lectures in
Mathematics. MR
0402720 (53 #6534)
 [Ar]
Shôrô
Araki, Typical formal groups in complex cobordism and
𝐾theory, Kinokuniya BookStore Co., Ltd., Tokyo, 1973.
Lectures in Mathematics, Department of Mathematics, Kyoto University, No.
6. MR
0375354 (51 #11549)
 [Ba]
Nils
Andreas Baas, On bordism theory of manifolds with
singularities, Math. Scand. 33 (1973), 279–302
(1974). MR
0346824 (49 #11547b)
 [BM]
Nils
Andreas Baas and Ib
Madsen, On the realization of certain modules over the Steenrod
algebra, Math. Scand. 31 (1972), 220–224. MR 0377879
(51 #14048)
 [H]
Michiel
Hazewinkel, Formal groups and applications, Pure and Applied
Mathematics, vol. 78, Academic Press, Inc. [Harcourt Brace Jovanovich,
Publishers], New YorkLondon, 1978. MR 506881
(82a:14020)
 [JW1]
David
Copeland Johnson and W.
Stephen Wilson, Projective dimension and BrownPeterson
homology, Topology 12 (1973), 327–353. MR 0334257
(48 #12576)
 [JW2]
David
Copeland Johnson and W.
Stephen Wilson, 𝐵𝑃 operations and Morava’s
extraordinary 𝐾theories, Math. Z. 144
(1975), no. 1, 55–75. MR 0377856
(51 #14025)
 [JW3]
David
Copeland Johnson and W.
Stephen Wilson, The projective dimension of the complex bordism of
EilenbergMacLane spaces, Osaka J. Math. 14 (1977),
no. 3, 533–536. MR 0467731
(57 #7584)
 [JW4]
David
Copeland Johnson and W.
Stephen Wilson, The BrownPeterson homology of elementary
𝑝groups, Amer. J. Math. 107 (1985),
no. 2, 427–453. MR 784291
(86j:55008), http://dx.doi.org/10.2307/2374422
 [JY]
David
Copeland Johnson and Zenichi
Yosimura, Torsion in BrownPeterson homology and Hurewicz
homomorphisms, Osaka J. Math. 17 (1980), no. 1,
117–136. MR
558323 (81b:55010)
 [L]
Peter
S. Landweber, Coherence, flatness and cobordism of classifying
spaces, Proc. Advanced Study Inst. on Algebraic Topology (Aarhus,
1970) Mat. Inst., Aarhus Univ., Aarhus, 1970, pp. 256–269. MR 0271964
(42 #6845)
 [M1]
John
Milnor, The Steenrod algebra and its dual, Ann. of Math. (2)
67 (1958), 150–171. MR 0099653
(20 #6092)
 [M2]
J.
Milnor, On the cobordism ring Ω* and a complex analogue.
I, Amer. J. Math. 82 (1960), 505–521. MR 0119209
(22 #9975)
 [M3]
J.
Milnor, On axiomatic homology theory, Pacific J. Math.
12 (1962), 337–341. MR 0159327
(28 #2544)
 [Mo]
Jack
Morava, A product for the oddprimary bordism of manifolds with
singularities, Topology 18 (1979), no. 3,
177–186. MR
546788 (80k:57063), http://dx.doi.org/10.1016/00409383(79)900016
 [Q]
Daniel
Quillen, On the formal group laws of unoriented
and complex cobordism theory, Bull. Amer. Math.
Soc. 75 (1969),
1293–1298. MR 0253350
(40 #6565), http://dx.doi.org/10.1090/S000299041969124018
 [R1]
Douglas
C. Ravenel, Complex cobordism and stable homotopy groups of
spheres, Pure and Applied Mathematics, vol. 121, Academic Press,
Inc., Orlando, FL, 1986. MR 860042
(87j:55003)
 [R2]
Douglas
C. Ravenel, Nilpotence and periodicity in stable homotopy
theory, Annals of Mathematics Studies, vol. 128, Princeton
University Press, Princeton, NJ, 1992. Appendix C by Jeff Smith. MR 1192553
(94b:55015)
 [RW1]
Douglas
C. Ravenel and W.
Stephen Wilson, The Hopf ring for complex cobordism, J. Pure
Appl. Algebra 9 (1976/77), no. 3, 241–280. MR 0448337
(56 #6644)
 [RW2]
Douglas
C. Ravenel and W.
Stephen Wilson, The Morava 𝐾theories of EilenbergMac Lane
spaces and the ConnerFloyd conjecture, Amer. J. Math.
102 (1980), no. 4, 691–748. MR 584466
(81i:55005), http://dx.doi.org/10.2307/2374093
 [RWY]
D. C. Ravenel, W. S. Wilson, and N. Yagita, BrownPeterson cohomology from Morava theory, theory 15 (1998), 147199. CMP 99:02
 [S]
Kathleen
Sinkinson, The cohomology of certain spectra associated with the
BrownPeterson spectrum, Duke Math. J. 43 (1976),
no. 3, 605–622. MR 0410725
(53 #14471)
 [T1]
Hirotaka
Tamanoi, The image of the BP Thom map for
EilenbergMac Lane spaces, Trans. Amer. Math.
Soc. 349 (1997), no. 3, 1209–1237. MR 1401530
(97i:55012), http://dx.doi.org/10.1090/S0002994797018266
 [T2]
H. Tamanoi, subalgebras, Milnor basis, and cohomology of EilenbergMac Lane spaces, To appear in J. Pure and Applied Algebra.
 [W]
W.
Stephen Wilson, The Ωspectrum for BrownPeterson cohomology.
I, Comment. Math. Helv. 48 (1973), 45–55;
corrigendum, ibid. 48 (1973), 194. MR 0326712
(48 #5055)
W.
Stephen Wilson, The Ωspectrum for BrownPeterson cohomology.
II, Amer. J. Math. 97 (1975), 101–123. MR 0383390
(52 #4271)
 [Y1]
Nobuaki
Yagita, On relations between BrownPeterson cohomology and the
ordinary mod 𝑝 cohomology theory, Kodai Math. J.
7 (1984), no. 2, 273–285. MR 744140
(85g:55007), http://dx.doi.org/10.2996/kmj/1138036912
 [Y2]
Nobuaki
Yagita, On the image
𝜌(𝐵𝑃*(𝑋)→𝐻*(𝑋;𝑍_{𝑝})),
Homotopy theory and related topics (Kyoto, 1984) Adv. Stud. Pure Math.,
vol. 9, NorthHolland, Amsterdam, 1987, pp. 335–344. MR 896964
(88j:55005)
 [Z]
Raphael
Zahler, The AdamsNovikov spectral sequence for the spheres,
Ann. of Math. (2) 96 (1972), 480–504. MR 0319197
(47 #7742)
 [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 Ktheory, 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), 279302. MR 49:11547b
 [BM]
 N. A. Baas and Ib Madsen, On the realization of certain modules over the Steenrod algebra, Math. Scand. 31 (1971), 220224. 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 BrownPeterson homology, Topology 12 (1973), 327353. MR 48:12576
 [JW2]
 D. C. Johnson and W. S. Wilson, BP operations and Morava's extraordinary Ktheories, Math. Z. 144 (1975), 5575. MR 51:14025
 [JW3]
 D. C. Johnson and W. S. Wilson, The projective dimension of the complex cobordism of EilenbergMacLane spaces, Osaka J. Math. 14 (1977), 533536. MR 57:7584
 [JW4]
 D. C. Johnson and W. S. Wilson, The BrownPeterson homology of elementary groups, Amer. J. Math. 107 (1985), 427453. MR 86j:55008
 [JY]
 D. C. Johnson and Z. Yoshimura, Torsion in BrownPeterson homology and Hurewicz homomorphisms, Osaka J. Math. 17 (1980), 117136. 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. 256269. MR 42:6845
 [M1]
 J. W. Milnor, The Steenrod algebra and its dual, Ann. of Math. 67 (1958), 150171. MR 20:6092
 [M2]
 J. W. Milnor, On the cobordism ring and a complex analogue, Amer. J. Math. 82 (1960), 505521. MR 22:9975
 [M3]
 J. W. Milnor, On axiomatic homology theory, Pacific J. Math. 12 (1962), 337341. MR 28:2544
 [Mo]
 J. Morava, A product for the oddprimary bordism of manifolds with singularities, Topology 18 (1979), 177186. MR 80k:57063
 [Q]
 D. Quillen, On the formal group laws of unoriented and complex cobordism theory, Bull. Amer. Math. Soc. 75 (1969), 12931298. 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), 241280. MR 56:6644
 [RW2]
 D. C. Ravenel and W. S. Wilson, The Morava K theories of EilenbergMacLane spaces and the ConnerFloyd conjecture, Amer. J. Math. 102 (1980), 691748. MR 81i:55005
 [RWY]
 D. C. Ravenel, W. S. Wilson, and N. Yagita, BrownPeterson cohomology from Morava theory, theory 15 (1998), 147199. CMP 99:02
 [S]
 K. Sinkinson, The cohomology of certain spectra associated with th BrownPeterson spectrum, Duke Math. J. 43 (1976), 605622. MR 53:14471
 [T1]
 H. Tamanoi, The image of the BP Thom map for EilenbergMac Lane spaces, Trans. AMS 349 (1997), 12091237. MR 97i:55012
 [T2]
 H. Tamanoi, subalgebras, Milnor basis, and cohomology of EilenbergMac Lane spaces, To appear in J. Pure and Applied Algebra.
 [W]
 W. S. Wilson, The spectrum for BrownPeterson cohomology, Part I, Comment. Math. Helv. 48 (1973), 4555; Part II, Amer. J. Math. 97 (1975), 101123.MR 48:5055; MR 52:4271
 [Y1]
 N. Yagita, On relations between BrownPeterson cohomology and the ordinary mod cohomology theory, Kodai Math. J. 7 (1984), 273285. MR 85g:55007
 [Y2]
 N. Yagita, On the image , Advanced Studies in Pure Math. 9 (1986, Homotopy Theory and Related Topics), 335344. MR 88j:55005
 [Z]
 R. Zahler, The AdamsNovikov spectral sequence for the spheres, Ann. of Math. 96 (1972), 480504. MR 47:7742
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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/S0002994799024848
PII:
S 00029947(99)024848
Keywords:
BrownPeterson (co)homology theory,
BP fundamental class,
BP topology,
EilenbergMac 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
