The image of the Thom map for Eilenberg-MacLane spaces

Author:
Hirotaka Tamanoi

Journal:
Trans. Amer. Math. Soc. **349** (1997), 1209-1237

MSC (1991):
Primary 55N22, 55P20, 55S25

DOI:
https://doi.org/10.1090/S0002-9947-97-01826-6

MathSciNet review:
1401530

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Abstract: Fundamental classes in cohomology of Eilenberg-MacLane spaces are defined. The image of the Thom map from cohomology to mod- cohomology is determined for arbitrary Eilenberg-MacLane 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- cohomology. This subalgebra in mod cohomology is invariant under the action of the Steenrod algebra, and it is annihilated by all Milnor primitives. We also show that cohomology determines Morava cohomology for Eilenberg-MacLane spaces.

**[A1]**J. F. Adams,*Lectures on Generalised Cohomology*, Lecture Notes in Math., vol. 99, Springer-Verlag, New York, 1969. MR**40:4943****[A2]**J. F. Adams,*Stable Homotopy and Generalized Homology*, University of Chicago Press, Chicago, Ill., 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****[BP]**E. H. Brown and F. P. Peterson,*A spectrum whose cohomology is the algebra of reduced -th powers*, Topology**5**(1966), 149-154. MR**33:719****[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 -groups*, Amer. J. Math.**107**(1985), 427-453. MR**86j:55008****[JY]**D. C. Johnson and Z. Yosimura,*Torsion in Brown-Peterson homology and Hurewicz homomorphisms*, Osaka J. Math.**17**(1980), 117-136. MR**81b:55010****[K]**D. Kraines,*On excess in the Milnor basis*, Bull. London Math. Soc.**3**(1971), 363-365. MR**45:9317****[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 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****[SY]**N. Shimada and N. Yagita,*Multiplications in the complex bordism theory with singularities*, Publ. R. I. M. S., Kyoto Univ.**12**(1976), 259-293. MR**54:3723****[T]**H. Tamanoi,*A decomposition formula for Milnor's Steenrod reduced powers, mod- cohomology of Eilenberg-MacLane spaces in terms of Milnor basis, and -subalgebras*, IHES preprint, IHES/M/95/51.**[W]**W. S. Wilson,*The -spectrum for Brown-Peterson cohomology*, Part I, Comment. Math. Helv.**48**(1973), 45-55;*Part II*, Amer. J. Math.**97**(1975), 101-123. MR**48:5505**; MR**52:4271****[Wu]**U. Würgler,*On products in a family of cohomology theories associated to the invariant prime ideals of*, Comment. Math. Helvet.**52**(1977), 457-481. MR**57:17624****[Y]**N. Yagita,*On the image*, Homotopy Theory and Related Topics (Kyoto, 1984), Adv. Stud. Pure Math., vol. 9, North-Holland, Amsterdam, 1987, pp. 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**

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Additional Information

**Hirotaka Tamanoi**

Affiliation:
Institut des Hautes Études Scientifiques, 35 Route de Chartres, 91440 Bures-sur-Yvette, France

Address at time of publication:
Department of Mathematics, University of California at Santa Cruz, Santa Cruz, California 95064

Email:
tamanoi@cats.ucsc.edu

DOI:
https://doi.org/10.1090/S0002-9947-97-01826-6

Keywords:
$BP$ cohomology theory,
$BP$ fundamental class,
Eilenberg--Mac Lane spaces,
Milnor primitives,
Morava $K$ theory,
Steenrod algebra,
Thom map

Received by editor(s):
October 5, 1995

Article copyright:
© Copyright 1997
American Mathematical Society