A Stong-Hattori spectral sequence

Johnson, David Copeland

doi:10.1090/S0002-9947-1973-0368040-7

A Stong-Hattori spectral sequence
HTML articles powered by AMS MathViewer

by David Copeland Johnson PDF

Trans. Amer. Math. Soc. 179 (1973), 211-225 Request permission

Abstract:

Let ${G_ \ast }(\;)$ be the Adams summand of connective K-theory localized at the prime p. Let $B{P_\ast }(\;)$ be Brown-Peterson homology for that prime. A spectral sequence is constructed with ${E^2}$ term determined by ${G_ \ast }(X)$ and whose ${E^\infty }$ terms give the quotients of a filtration of $B{P_ \ast }(X)$ where X is a connected spectrum. A torsion property of the differentials implies the Stong-Hattori theorem.

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

Quillen’s work on formal groups and complex cobordism

J. Frank Adams and Arunas Liulevicius, The Hurewicz homomorphism for $\textrm {MU}$ and $\textrm {BP}$, J. London Math. Soc. (2) 5 (1972), 539–545. MR 353309, DOI 10.1112/jlms/s2-5.3.539
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
Edgar H. Brown Jr., Cohomology theories, Ann. of Math. (2) 75 (1962), 467–484. MR 138104, DOI 10.2307/1970209
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
Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. MR 0077480
P. E. Conner and E. E. Floyd, The relation of cobordism to $K$-theories, Lecture Notes in Mathematics, No. 28, Springer-Verlag, Berlin-New York, 1966. MR 0216511, DOI 10.1007/BFb0071091
P. E. Conner and Larry Smith, On the complex bordism of finite complexes, Inst. Hautes Études Sci. Publ. Math. 37 (1969), 117–221. MR 267571, DOI 10.1007/BF02684888
P. E. Conner and Larry Smith, On the complex bordism of finite complexes. II, J. Differential Geometry 6 (1971/72), 135–174. MR 298672, DOI 10.4310/jdg/1214430402
Albrecht Dold, On general cohomology. Chapters 1–9, Aarhus Universitet, Matematisk Institut, Aarhus, 1968. Lectures given at the Nordic Summer School in Mathematics (June 16th-July 6th, 1968). MR 0254838
Akio Hattori, Integral characteristic numbers for weakly almost complex manifolds, Topology 5 (1966), 259–280. MR 192517, DOI 10.1016/0040-9383(66)90010-3

Constructing formal groups

Over

algebras

Cobordism and connective K-theory

David Copeland Johnson and Larry Smith, On the relation of complex cobordism to connective $K$-theory, Proc. Cambridge Philos. Soc. 70 (1971), 19–22. MR 293619, DOI 10.1017/s0305004100049616
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
Peter S. Landweber, Complex bordism of classifying spaces, Proc. Amer. Math. Soc. 27 (1971), 175–179. MR 268885, DOI 10.1090/S0002-9939-1971-0268885-1

On the algebra

Saunders MacLane, Homology, 1st ed., Die Grundlehren der mathematischen Wissenschaften, Band 114, Springer-Verlag, Berlin-New York, 1967. MR 0349792
Larry Smith, A note on the Stong-Hattori theorem, Illinois J. Math. 17 (1973), 285–289. MR 322852
Larry Smith, On realizing complex bordism modules. Applications to the stable homotopy of spheres, Amer. J. Math. 92 (1970), 793–856. MR 275429, DOI 10.2307/2373397
Larry Smith, On the relation of connective $K$-theory to homology, Proc. Cambridge Philos. Soc. 68 (1970), 637–639. MR 263072, DOI 10.1017/s0305004100076623
R. E. Stong, Relations among characteristic numbers. I, Topology 4 (1965), 267–281. MR 192515, DOI 10.1016/0040-9383(65)90011-X
Rainer Vogt, Boardman’s stable homotopy category, Lecture Notes Series, No. 21, Aarhus Universitet, Matematisk Institut, Aarhus, 1970. MR 0275431
Raphael Zahler, The Adams-Novikov spectral sequence for the spheres, Ann. of Math. (2) 96 (1972), 480–504. MR 319197, DOI 10.2307/1970821