The image of the 𝐵𝑃 Thom map for Eilenberg-MacLane spaces

Tamanoi, Hirotaka

doi:10.1090/S0002-9947-97-01826-6

The image of the $BP$ Thom map for Eilenberg-MacLane spaces
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by Hirotaka Tamanoi PDF

Trans. Amer. Math. Soc. 349 (1997), 1209-1237 Request permission

Abstract:

Fundamental classes in $BP$ cohomology of Eilenberg–Mac Lane spaces are defined. The image of the Thom map from $BP$ cohomology to mod-$p$ cohomology is determined for arbitrary Eilenberg–Mac Lane spaces. This image is a polynomial subalgebra generated by infinitely many elements obtained by applying a maximum number of Milnor primitives to the fundamental class in mod-$p$ cohomology. This subalgebra in mod $p$ cohomology is invariant under the action of the Steenrod algebra, and it is annihilated by all Milnor primitives. We also show that $BP$ cohomology determines Morava $K$ cohomology for Eilenberg–Mac Lane spaces.

References

J. F. Adams, Lectures on generalised cohomology, Category Theory, Homology Theory and their Applications, III (Battelle Institute Conference, Seattle, Wash., 1968, Vol. Three), Springer, Berlin, 1969, pp. 1–138. MR 0251716
J. F. Adams, Stable homotopy and generalised homology, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, Ill.-London, 1974. MR 0402720
Shôrô Araki, Typical formal groups in complex cobordism and $K$-theory, Lectures in Mathematics, Department of Mathematics, Kyoto University, No. 6, Kinokuniya Book Store Co., Ltd., Tokyo, 1973. MR 0375354
Nils Andreas Baas, Bordism theories with singularities, Proceedings of the Advanced Study Institute on Algebraic Topology (Aarhus Univ., Aarhus, 1970) Various Publ. Ser., No. 13, Mat. Inst., Aarhus Univ., Aarhus, 1970, pp. 1–16. MR 0346823
Nils Andreas Baas and Ib Madsen, On the realization of certain modules over the Steenrod algebra, Math. Scand. 31 (1972), 220–224. MR 377879, DOI 10.7146/math.scand.a-11427
Edgar H. Brown Jr. and Franklin P. Peterson, A spectrum whose $Z_{p}$ cohomology is the algebra of reduced $p^{th}$ powers, Topology 5 (1966), 149–154. MR 192494, DOI 10.1016/0040-9383(66)90015-2
Michiel Hazewinkel, Formal groups and applications, Pure and Applied Mathematics, vol. 78, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 506881
David Copeland Johnson and W. Stephen Wilson, Projective dimension and Brown-Peterson homology, Topology 12 (1973), 327–353. MR 334257, DOI 10.1016/0040-9383(73)90027-X
David Copeland Johnson and W. Stephen Wilson, $BP$ operations and Morava’s extraordinary $K$-theories, Math. Z. 144 (1975), no. 1, 55–75. MR 377856, DOI 10.1007/BF01214408
David Copeland Johnson and W. Stephen Wilson, The projective dimension of the complex bordism of Eilenberg-MacLane spaces, Osaka Math. J. 14 (1977), no. 3, 533–536. MR 467731
David Copeland Johnson and W. Stephen Wilson, The Brown-Peterson homology of elementary $p$-groups, Amer. J. Math. 107 (1985), no. 2, 427–453. MR 784291, DOI 10.2307/2374422
David Copeland Johnson and Zen-ichi Yosimura, Torsion in Brown-Peterson homology and Hurewicz homomorphisms, Osaka Math. J. 17 (1980), no. 1, 117–136. MR 558323
D. Kraines, On excess in the Milnor basis, Bull. London Math. Soc. 3 (1971), 363–365. MR 300271, DOI 10.1112/blms/3.3.363
John Milnor, The Steenrod algebra and its dual, Ann. of Math. (2) 67 (1958), 150–171. MR 99653, DOI 10.2307/1969932
J. Milnor, On the cobordism ring $\Omega ^{\ast }$ and a complex analogue. I, Amer. J. Math. 82 (1960), 505–521. MR 119209, DOI 10.2307/2372970
J. Milnor, On axiomatic homology theory, Pacific J. Math. 12 (1962), 337–341. MR 159327, DOI 10.2140/pjm.1962.12.337
Jack Morava, A product for the odd-primary bordism of manifolds with singularities, Topology 18 (1979), no. 3, 177–186. MR 546788, DOI 10.1016/0040-9383(79)90001-6
Daniel Quillen, On the formal group laws of unoriented and complex cobordism theory, Bull. Amer. Math. Soc. 75 (1969), 1293–1298. MR 253350, DOI 10.1090/S0002-9904-1969-12401-8
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
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
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 448337, DOI 10.1016/0022-4049(77)90070-6
Douglas C. Ravenel and W. Stephen Wilson, The Morava $K$-theories of Eilenberg-Mac Lane spaces and the Conner-Floyd conjecture, Amer. J. Math. 102 (1980), no. 4, 691–748. MR 584466, DOI 10.2307/2374093
Nobuo Shimada and Nobuaki Yagita, Multiplications in the complex bordism theory with singularities, Publ. Res. Inst. Math. Sci. 12 (1976/77), no. 1, 259–293. MR 0415642, DOI 10.2977/prims/1195190968
H. Tamanoi, A decomposition formula for Milnor’s Steenrod reduced powers, mod-$p$ cohomology of Eilenberg–Mac Lane spaces in terms of Milnor basis, and $Q$-subalgebras, IHES preprint, IHES/M/95/51.
B. Vujanović, Dj. Djukić, and M. Pavlović, A variational principle for the laminar boundary layer theory, Boll. Un. Mat. Ital. (4) 7 (1973), 377–391 (English, with Italian summary). MR 0327163
Urs Würgler, On products in a family of cohomology theories associated to the invariant prime ideals of $\pi _\ast (\textrm {BP})$, Comment. Math. Helv. 52 (1977), no. 4, 457–481. MR 478135, DOI 10.1007/BF02567379
Nobuaki Yagita, On the image $\rho (\textrm {BP}^*(X)\to H^*(X;Z_p))$, Homotopy theory and related topics (Kyoto, 1984) Adv. Stud. Pure Math., vol. 9, North-Holland, Amsterdam, 1987, pp. 335–344. MR 896964, DOI 10.2969/aspm/00910335
Raphael Zahler, The Adams-Novikov spectral sequence for the spheres, Ann. of Math. (2) 96 (1972), 480–504. MR 319197, DOI 10.2307/1970821