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The cohomology of finite H-spaces as U(M) algebras: I

Published online by Cambridge University Press:  24 October 2008

Richard Kane
Richard Kane
Affiliation:
University of Western Ontario, London, Ontario
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By an H-space (X, μ) we will mean a topological space X having the homotopy type of a connected CW complex of finite type together with a basepoint preserving map μ: x × X → X with two sided homotopy unit. Let p be a prime and let /p be the integers reduced modp. Given an H-space (X, μ) then H*(X;/p) is a commutative associative Hopf algebra over the Steenrod algebra A*(p) and H*(X;/p) is iso- morphic, as an algebra, to a tensor product [⊗ , where each algebra At is generated by a single element ai (see Theorem 7.11 of (24)). The decomposition Ai is called a Borel decomposition and the elements {ai} are called the Borel generators of the decomposition. The decomposition ⊗ Ai and the resulting generators {at} are far from unique. Many choices are possible. Since A*(p) acts on H*(X;/p) an obvious restriction would be to choose the Borel decomposition to be compatible with this action. We would like the /p module generated by the Borel generators and their iterated pth powers to be invariant under the action of A * (p). More precisely we would like H*(X;/p) to be the enveloping algebra U(M) of an unstable Steenrod module M (see § 2). If H*(X;/p) admits such a choice then it is called a U(M) algebra. The fact that H*(X; /p) is a U(M) algebra has applications in homotopy theory. In particular there exist unstable Adams spectral sequences which can be used to calculate the homotopy groups of X (see (21) and (7)). However, the question of U(M) structures for modp cohomology seems of most interest simply as a classification device for finite H-spaces.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 89 , Issue 3 , May 1981 , pp. 473 - 490
Copyright
Copyright © Cambridge Philosophical Society 1981

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References

REFERENCES

(1)Adams, J. P.Stable homotopy and generalized homology (University of Chicago Press, 1974).Google Scholar
(2)Baum, P. and Browder, W.The cohomology of quotients of classical groups. Topology 3 (1965), 305–336.Google Scholar
(3)Browder, W.The cohomology of covering spaces of H-spaces. Bull. Amer. Math. Soc. 65 (1959), 140–141.CrossRefGoogle Scholar
(4)Browder, W.Torsion in H-Spaces. Annals of Math. 74 (1961), 24–51.Google Scholar
(5)Browder, W.On differential Hopf algebras. Trans. Amer. Math. Soc. 107 (1963), 153–176.CrossRefGoogle Scholar
(6)Clark, A.Homotopy commutativity and the Moore spectral sequence. Pacific J. Math. 15 (1965), 419–426.CrossRefGoogle Scholar
(7)Harper, J.Homotopy groups of H-Spaces. I. Comm. Math. Helv. 47 (1972), 311–331.Google Scholar
(8)Hazewtnkel, M.A universal formal group nad complex cobordism. Bull. Amer. Math. Soc. 81 (1975), 930–933.CrossRefGoogle Scholar
(9)Johnson, D. C. and Wilson, W. S.Projective dimension and Brown-Peterson homology. Topology 12 (1973), 327–353.CrossRefGoogle Scholar
(10)Kane, R.Loop Spaces without p torsion. Pacific J. Math. 60 (1975), 189–201.CrossRefGoogle Scholar
(11)Kane, R.The module of indecomposables for finite H-spaces. Trans. Amer. Math. Soc. 222 (1976), 303–318.Google Scholar
(12)Kane, R.The BP homology of H-spaces. Trans. Amer. Math. Soc. 241 (1978), 99–120.Google Scholar
(13)Kane, R.Rational BP operations and the chern character. Math. Proc. Cambridge Philos. Soc. 84 (1978), 65–72.CrossRefGoogle Scholar
(14)Kane, R.The module of indecomposables for Mod 2 finite H-spaces. Trans. Amer. Soc. 249 (1979), 425–433.Google Scholar
(15)Kane, R.Brown-Peterson operations and Steenrod modules. Quart. J. Math. Oxford 30 (1979), 455–467.CrossRefGoogle Scholar
(16)Kane, R. The cohomology of finite H-spaces as U(M) algebras: II. (To appear.)Google Scholar
(17)Landweber, P.Elements of infinite filtration in complex cobordism. Math. Scand. 30 (1972), 223–226.CrossRefGoogle Scholar
(18)Lin, J.Torsion in H-spaces: I. Annals of Math. 103 (1976), 457–487.Google Scholar
(19)Lin, J.Torsion in H-spaces: II. Annals of Math. 107 (1978), 41–88.CrossRefGoogle Scholar
(20)Lin, J.Even generators in the Mod 2 cohomology of a finite H-space. Math. Scand. 44 (1979), 295–312.CrossRefGoogle Scholar
(21)Massey, W. S. and PeterSon, F. P.The cohomology structure of certain fibre spaces. (Amer. Math. Soc. Memoirs (74), 1967).Google Scholar
(22)May, J. P. and Zabrodsky, A.H* (Spin (n)) as a Hopf algebra. J. Pure and Applied Algebra 10 (1977), 193–200.CrossRefGoogle Scholar
(23)Milnor, J.The Steenrod algebra and its dual. Annals of Math. 67 (1958), 150–171.CrossRefGoogle Scholar
(24)Milnor, J. and Moore, J. C.On the structure of Hopf algebra. Annals of Math. 81 (1965), 211–264.Google Scholar
(25)Moore, J. C.La Suspension. Séminaire H. Cartan-J. C. Moore 12 (1959/1960), Exposé 6.Google Scholar
(26)Quillen, D. G.On the formal group laws of unoriented and complex cobordism theory. Bull. Amer. Math. Soc. 75 (1969), 1293–1298.CrossRefGoogle Scholar
(27)Rector, D. L.Steenrod Operations in the Eilenberg-Moore Spectral Sequence. Comrn. Math. Helv. 45 (1970), 540–542.CrossRefGoogle Scholar
(28)Smith, L.On the Eilenberg-Moore Spectral Sequence. Proc. Sympos. Pure Math. 22, Amer. Math. Soc. (1971), 231–241.Google Scholar
(29)Wall, C. T. C.Finiteness conditions for CW-complexes. Annals of Math. 81 (1965), 56–69.CrossRefGoogle Scholar
(30)Zabrodsky, A.Secondary cohomology in the module of indecomposables. Algebraic Topology Conference, Aarhus, 1970.Google Scholar
(31)Zabrodsky, A.Some relations in the mod 3 cohomology of H-spaces. Israel J. Math. 33 (1979), 59–72.Google Scholar
(32)Zahler, R.The Adams-Novikov spectral sequence for the sphere. Annals of Math. 96 (1972), 480–504.CrossRefGoogle Scholar