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Rational BP operations and the Chern character

Published online by Cambridge University Press:  24 October 2008

Richard Kane
Richard Kane
Affiliation:
University of Alberta
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Extract

This paper will study certain rational BP operations. We will work in the category of spaces having the homotopy type of CW complexes. Recall that for − 1 ≤ n ≤ ∞ there is defined in this category the homology theory BP(n)*(X). (See (7).) In particular, BP(∞)*(X) = BP*(X) is the usual Brown Peterson homology of X, BP(1)*(X) = l*(X) is the canonical direct summand of connective K-theory with Qp coefficients (Qp are the integers localized at the prime p), BP(0)*(X) = Hℤ*(X) is ordinary homology with Qp coefficients, and BP(−1)*(X) = Hℤ/p*(X) is ordinary homology with ℤp coefficients (ℤp are the integers reduced mod p). For 1 ≤ n ≤ ∞, BP(n)*(X) is a module over BP(n)*(pt) = Qp[v1, …, vn] (deg vs = 2ps − 2). The BP(n) theories are all related via canonical maps p(m, n):BP(m)*(X) → BP(n)*(X) whose performance is described in (7).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 84 , Issue 1 , July 1978 , pp. 65 - 72
Copyright
Copyright © Cambridge Philosophical Society 1978

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References

REFERENCES

(1)Adams, J. F.On Chern characters and the structure of the unitary group. Proc. Cambridge Philos. Soc. 57 (1961), 189–199.CrossRefGoogle Scholar
(2)Adams, J. F.Chern characters revisited. Illinois J. Math. 17 (1973), 333–336.CrossRefGoogle Scholar
(3)Adams, J. F.Lectures on generalized cohomology. Lecture Notes in Mathematics 99 (1969), 1–138.CrossRefGoogle Scholar
(4)Adams, J. F.Stable homotopy and generalized homology (University of Chicago Press, 1974).Google Scholar
(5)Dold, A.On general cohomology. Aarhus University Lecture Notes (1968).Google Scholar
(6)Hazewinkel, M.A universal formal group and complex cobordism. Bull. Amer. Math. Soc. 81 (1975), 930–933.CrossRefGoogle Scholar
(7)Johnson, D. C. And Wilson, W. S.Projective dimension and Brown-Peterson homology. Topology 12 (1973), 327–353.CrossRefGoogle Scholar
(8)Milnor, J.The Steenrod algebra and its dual. Ann. of Math. 67 (1958), 150–171.CrossRefGoogle Scholar
(9)Milnor, J. And Moore, J. C.On the structure of Hopf algebras. Ann. of Math. 81 (1965), 211–264.CrossRefGoogle Scholar
(10)Quillen, D. G.On the formal group laws of unoriented and complex cobordism theory. Bull. Amer. Math. Soc. 75 (1969), 1293–1298.CrossRefGoogle Scholar
(11)Smith, L.The Todd character and the integrality theorem for the Chern character. Illinois J. Math. 17 (1973), 301–310.CrossRefGoogle Scholar
(12)Zahler, R.The Adams-Novikov spectral sequence for the spheres. Ann. of Math. 9 (1972), 480–504.CrossRefGoogle Scholar